\begin{patch}{ugxProblemSeriesBernoulliPagePatch1}
\begin{paste}{ugxProblemSeriesBernoulliPageFull1}{ugxProblemSeriesBernoulliPageEmpty1}
\pastebutton{ugxProblemSeriesBernoulliPageFull1}{\hidepaste}
\tab{5}\spadcommand{reduce(+,[m**4 for m in 1..10])}
\indentrel{3}\begin{verbatim}
   (1)  25333
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty1}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty1}{ugxProblemSeriesBernoulliPagePatch1}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty1}{\showpaste}
\tab{5}\spadcommand{reduce(+,[m**4 for m in 1..10])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch2}
\begin{paste}{ugxProblemSeriesBernoulliPageFull2}{ugxProblemSeriesBernoulliPageEmpty2}
\pastebutton{ugxProblemSeriesBernoulliPageFull2}{\hidepaste}
\tab{5}\spadcommand{sum4 := sum(m**4, m = 1..k)\bound{sum4 }}
\indentrel{3}\begin{verbatim}
          5      4      3
        6k  + 15k  + 10k  - k
   (2)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                  30
                      Type: Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty2}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty2}{ugxProblemSeriesBernoulliPagePatch2}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty2}{\showpaste}
\tab{5}\spadcommand{sum4 := sum(m**4, m = 1..k)\bound{sum4 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch3}
\begin{paste}{ugxProblemSeriesBernoulliPageFull3}{ugxProblemSeriesBernoulliPageEmpty3}
\pastebutton{ugxProblemSeriesBernoulliPageFull3}{\hidepaste}
\tab{5}\spadcommand{eval(sum4, k = 10)\free{sum4 }}
\indentrel{3}\begin{verbatim}
   (3)  25333
                      Type: Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty3}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty3}{ugxProblemSeriesBernoulliPagePatch3}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty3}{\showpaste}
\tab{5}\spadcommand{eval(sum4, k = 10)\free{sum4 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch4}
\begin{paste}{ugxProblemSeriesBernoulliPageFull4}{ugxProblemSeriesBernoulliPageEmpty4}
\pastebutton{ugxProblemSeriesBernoulliPageFull4}{\hidepaste}
\tab{5}\spadcommand{f := t*exp(x*t) / (exp(t) - 1)\bound{f }}
\indentrel{3}\begin{verbatim}
            t x
        t %e
   (4)  ÄÄÄÄÄÄÄ
          t
        %e  - 1
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty4}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty4}{ugxProblemSeriesBernoulliPagePatch4}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty4}{\showpaste}
\tab{5}\spadcommand{f := t*exp(x*t) / (exp(t) - 1)\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch5}
\begin{paste}{ugxProblemSeriesBernoulliPageFull5}{ugxProblemSeriesBernoulliPageEmpty5}
\pastebutton{ugxProblemSeriesBernoulliPageFull5}{\hidepaste}
\tab{5}\spadcommand{)set streams calculate 5\bound{set }}
\indentrel{3}\begin{verbatim}
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty5}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty5}{ugxProblemSeriesBernoulliPagePatch5}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty5}{\showpaste}
\tab{5}\spadcommand{)set streams calculate 5\bound{set }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch6}
\begin{paste}{ugxProblemSeriesBernoulliPageFull6}{ugxProblemSeriesBernoulliPageEmpty6}
\pastebutton{ugxProblemSeriesBernoulliPageFull6}{\hidepaste}
\tab{5}\spadcommand{ff := taylor(f,t = 0)\free{f set }\bound{ff }}
\indentrel{3}\begin{verbatim}
   (5)
                      2                 3     2
         2x - 1     6x  - 6x + 1  2   2x  - 3x  + x  3
     1 + ÄÄÄÄÄÄ t + ÄÄÄÄÄÄÄÄÄÄÄÄ t  + ÄÄÄÄÄÄÄÄÄÄÄÄÄ t
            2            12                 12
   + 
        4      3      2            5      4      3
     30x  - 60x  + 30x  - 1  4   6x  - 15x  + 10x  - x  5
     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t  + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
               720                        720
   + 
        6
     O(t )
   Type: UnivariateTaylorSeries(Expression Integer,t,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty6}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty6}{ugxProblemSeriesBernoulliPagePatch6}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty6}{\showpaste}
\tab{5}\spadcommand{ff := taylor(f,t = 0)\free{f set }\bound{ff }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch7}
\begin{paste}{ugxProblemSeriesBernoulliPageFull7}{ugxProblemSeriesBernoulliPageEmpty7}
\pastebutton{ugxProblemSeriesBernoulliPageFull7}{\hidepaste}
\tab{5}\spadcommand{factorial(6) * coefficient(ff,6)\free{ff }}
\indentrel{3}\begin{verbatim}
           6       5       4      2
        42x  - 126x  + 105x  - 21x  + 1
   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                       42
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty7}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty7}{ugxProblemSeriesBernoulliPagePatch7}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty7}{\showpaste}
\tab{5}\spadcommand{factorial(6) * coefficient(ff,6)\free{ff }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch8}
\begin{paste}{ugxProblemSeriesBernoulliPageFull8}{ugxProblemSeriesBernoulliPageEmpty8}
\pastebutton{ugxProblemSeriesBernoulliPageFull8}{\hidepaste}
\tab{5}\spadcommand{g := eval(f, x = x + 1) - f\bound{g }\free{f }}
\indentrel{3}\begin{verbatim}
            t x + t       t x
        t %e        - t %e
   (7)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                 t
               %e  - 1
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty8}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty8}{ugxProblemSeriesBernoulliPagePatch8}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty8}{\showpaste}
\tab{5}\spadcommand{g := eval(f, x = x + 1) - f\bound{g }\free{f }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch9}
\begin{paste}{ugxProblemSeriesBernoulliPageFull9}{ugxProblemSeriesBernoulliPageEmpty9}
\pastebutton{ugxProblemSeriesBernoulliPageFull9}{\hidepaste}
\tab{5}\spadcommand{normalize(g)\free{g }}
\indentrel{3}\begin{verbatim}
            t x
   (8)  t %e
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty9}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty9}{ugxProblemSeriesBernoulliPagePatch9}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty9}{\showpaste}
\tab{5}\spadcommand{normalize(g)\free{g }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch10}
\begin{paste}{ugxProblemSeriesBernoulliPageFull10}{ugxProblemSeriesBernoulliPageEmpty10}
\pastebutton{ugxProblemSeriesBernoulliPageFull10}{\hidepaste}
\tab{5}\spadcommand{taylor(g,t = 0)\free{g }}
\indentrel{3}\begin{verbatim}
                    2       3       4
               2   x   3   x   4   x   5      6
   (9)  t + x t  + ÄÄ t  + ÄÄ t  + ÄÄ t  + O(t )
                    2       6      24
   Type: UnivariateTaylorSeries(Expression Integer,t,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty10}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty10}{ugxProblemSeriesBernoulliPagePatch10}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty10}{\showpaste}
\tab{5}\spadcommand{taylor(g,t = 0)\free{g }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch11}
\begin{paste}{ugxProblemSeriesBernoulliPageFull11}{ugxProblemSeriesBernoulliPageEmpty11}
\pastebutton{ugxProblemSeriesBernoulliPageFull11}{\hidepaste}
\tab{5}\spadcommand{B5 := factorial(5) * coefficient(ff,5)\free{ff }\bound{B5 }}
\indentrel{3}\begin{verbatim}
           5      4      3
         6x  - 15x  + 10x  - x
   (10)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                   6
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty11}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty11}{ugxProblemSeriesBernoulliPagePatch11}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty11}{\showpaste}
\tab{5}\spadcommand{B5 := factorial(5) * coefficient(ff,5)\free{ff }\bound{B5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch12}
\begin{paste}{ugxProblemSeriesBernoulliPageFull12}{ugxProblemSeriesBernoulliPageEmpty12}
\pastebutton{ugxProblemSeriesBernoulliPageFull12}{\hidepaste}
\tab{5}\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1))\free{B5 }}
\indentrel{3}\begin{verbatim}
           5      4      3
         6k  + 15k  + 10k  - k
   (11)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                   30
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty12}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty12}{ugxProblemSeriesBernoulliPagePatch12}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty12}{\showpaste}
\tab{5}\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1))\free{B5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPagePatch13}
\begin{paste}{ugxProblemSeriesBernoulliPageFull13}{ugxProblemSeriesBernoulliPageEmpty13}
\pastebutton{ugxProblemSeriesBernoulliPageFull13}{\hidepaste}
\tab{5}\spadcommand{sum4\free{sum4 }}
\indentrel{3}\begin{verbatim}
           5      4      3
         6k  + 15k  + 10k  - k
   (12)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                   30
                      Type: Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesBernoulliPageEmpty13}
\begin{paste}{ugxProblemSeriesBernoulliPageEmpty13}{ugxProblemSeriesBernoulliPagePatch13}
\pastebutton{ugxProblemSeriesBernoulliPageEmpty13}{\showpaste}
\tab{5}\spadcommand{sum4\free{sum4 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePagePatch1}
\begin{paste}{ugProblemLaplacePageFull1}{ugProblemLaplacePageEmpty1}
\pastebutton{ugProblemLaplacePageFull1}{\hidepaste}
\tab{5}\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)}
\indentrel{3}\begin{verbatim}
             3
           4a
   (1)  ÄÄÄÄÄÄÄÄ
         4     4
        s  + 4a
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePageEmpty1}
\begin{paste}{ugProblemLaplacePageEmpty1}{ugProblemLaplacePagePatch1}
\pastebutton{ugProblemLaplacePageEmpty1}{\showpaste}
\tab{5}\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePagePatch2}
\begin{paste}{ugProblemLaplacePageFull2}{ugProblemLaplacePageEmpty2}
\pastebutton{ugProblemLaplacePageFull2}{\hidepaste}
\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)}
\indentrel{3}\begin{verbatim}
   (2)  - log(s - a) + log(s - b)
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePageEmpty2}
\begin{paste}{ugProblemLaplacePageEmpty2}{ugProblemLaplacePagePatch2}
\pastebutton{ugProblemLaplacePageEmpty2}{\showpaste}
\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePagePatch3}
\begin{paste}{ugProblemLaplacePageFull3}{ugProblemLaplacePageEmpty3}
\pastebutton{ugProblemLaplacePageFull3}{\hidepaste}
\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)}
\indentrel{3}\begin{verbatim}
             2    2
   (3)  log(s  + a ) - 2log(s)
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePageEmpty3}
\begin{paste}{ugProblemLaplacePageEmpty3}{ugProblemLaplacePagePatch3}
\pastebutton{ugProblemLaplacePageEmpty3}{\showpaste}
\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePagePatch4}
\begin{paste}{ugProblemLaplacePageFull4}{ugProblemLaplacePageEmpty4}
\pastebutton{ugProblemLaplacePageFull4}{\hidepaste}
\tab{5}\spadcommand{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)}
\indentrel{3}\begin{verbatim}
                    1
   (4)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
           2             3    2
        b s  + 2a b s + b  + a b
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePageEmpty4}
\begin{paste}{ugProblemLaplacePageEmpty4}{ugProblemLaplacePagePatch4}
\pastebutton{ugProblemLaplacePageEmpty4}{\showpaste}
\tab{5}\spadcommand{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePagePatch5}
\begin{paste}{ugProblemLaplacePageFull5}{ugProblemLaplacePageEmpty5}
\pastebutton{ugProblemLaplacePageFull5}{\hidepaste}
\tab{5}\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)}
\indentrel{3}\begin{verbatim}
             2    2         2    2
        log(s  + b ) - log(s  + a )
   (5)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                     2
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePageEmpty5}
\begin{paste}{ugProblemLaplacePageEmpty5}{ugProblemLaplacePagePatch5}
\pastebutton{ugProblemLaplacePageEmpty5}{\showpaste}
\tab{5}\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePagePatch6}
\begin{paste}{ugProblemLaplacePageFull6}{ugProblemLaplacePageEmpty6}
\pastebutton{ugProblemLaplacePageFull6}{\hidepaste}
\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)}
\indentrel{3}\begin{verbatim}
          b    s + c - a
        %e log(ÄÄÄÄÄÄÄÄÄ)
                   c
   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
              s - a
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePageEmpty6}
\begin{paste}{ugProblemLaplacePageEmpty6}{ugProblemLaplacePagePatch6}
\pastebutton{ugProblemLaplacePageEmpty6}{\showpaste}
\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePagePatch7}
\begin{paste}{ugProblemLaplacePageFull7}{ugProblemLaplacePageEmpty7}
\pastebutton{ugProblemLaplacePageFull7}{\hidepaste}
\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
\indentrel{3}\begin{verbatim}
               2    2
              s  + b             d
        a log(ÄÄÄÄÄÄÄ) + 2c atan(Ä)
                  2              s
                 b
   (7)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                     2s
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePageEmpty7}
\begin{paste}{ugProblemLaplacePageEmpty7}{ugProblemLaplacePagePatch7}
\pastebutton{ugProblemLaplacePageEmpty7}{\showpaste}
\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePagePatch8}
\begin{paste}{ugProblemLaplacePageFull8}{ugProblemLaplacePageEmpty8}
\pastebutton{ugProblemLaplacePageFull8}{\hidepaste}
\tab{5}\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)}
\indentrel{3}\begin{verbatim}
                                    2
          4     2 2    4           t           3
        (s  + 2a s  + a )laplace(%e  ,t,s) + 2a
   (8)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                      4     2 2    4
                     s  + 2a s  + a
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLaplacePageEmpty8}
\begin{paste}{ugProblemLaplacePageEmpty8}{ugProblemLaplacePagePatch8}
\pastebutton{ugProblemLaplacePageEmpty8}{\showpaste}
\tab{5}\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch1}
\begin{paste}{ugxProblemSeriesFunctionsPageFull1}{ugxProblemSeriesFunctionsPageEmpty1}
\pastebutton{ugxProblemSeriesFunctionsPageFull1}{\hidepaste}
\tab{5}\spadcommand{x := series 'x\bound{x }}
\indentrel{3}\begin{verbatim}
   (1)  x
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty1}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty1}{ugxProblemSeriesFunctionsPagePatch1}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty1}{\showpaste}
\tab{5}\spadcommand{x := series 'x\bound{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch2}
\begin{paste}{ugxProblemSeriesFunctionsPageFull2}{ugxProblemSeriesFunctionsPageEmpty2}
\pastebutton{ugxProblemSeriesFunctionsPageFull2}{\hidepaste}
\tab{5}\spadcommand{rat := x**2 / (1 - 6*x + x**2)\free{x }\bound{rat }}
\indentrel{3}\begin{verbatim}
   (2)
      2     3      4       5        6        7         8
     x  + 6x  + 35x  + 204x  + 1189x  + 6930x  + 40391x
   + 
            9           10           11            12
     235416x  + 1372105x   + 7997214x   + 46611179x
   + 
        13
     O(x  )
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty2}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty2}{ugxProblemSeriesFunctionsPagePatch2}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty2}{\showpaste}
\tab{5}\spadcommand{rat := x**2 / (1 - 6*x + x**2)\free{x }\bound{rat }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch3}
\begin{paste}{ugxProblemSeriesFunctionsPageFull3}{ugxProblemSeriesFunctionsPageEmpty3}
\pastebutton{ugxProblemSeriesFunctionsPageFull3}{\hidepaste}
\tab{5}\spadcommand{sin(rat)\free{rat }}
\indentrel{3}\begin{verbatim}
   (3)
      2     3      4       5   7133  6        7   80711  8
     x  + 6x  + 35x  + 204x  + ÄÄÄÄ x  + 6927x  + ÄÄÄÄÄ x
                                 6                  2
   + 
            9   164285281  10   31888513  11
     235068x  + ÄÄÄÄÄÄÄÄÄ x   + ÄÄÄÄÄÄÄÄ x
                   120              4
   + 
     371324777  12      13
     ÄÄÄÄÄÄÄÄÄ x   + O(x  )
         8
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty3}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty3}{ugxProblemSeriesFunctionsPagePatch3}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty3}{\showpaste}
\tab{5}\spadcommand{sin(rat)\free{rat }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch4}
\begin{paste}{ugxProblemSeriesFunctionsPageFull4}{ugxProblemSeriesFunctionsPageEmpty4}
\pastebutton{ugxProblemSeriesFunctionsPageFull4}{\hidepaste}
\tab{5}\spadcommand{y : UTS(FRAC INT,y,0) := y\bound{y }}
\indentrel{3}\begin{verbatim}
   (4)  y
     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty4}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty4}{ugxProblemSeriesFunctionsPagePatch4}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty4}{\showpaste}
\tab{5}\spadcommand{y : UTS(FRAC INT,y,0) := y\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch5}
\begin{paste}{ugxProblemSeriesFunctionsPageFull5}{ugxProblemSeriesFunctionsPageEmpty5}
\pastebutton{ugxProblemSeriesFunctionsPageFull5}{\hidepaste}
\tab{5}\spadcommand{exp(y)\free{y }}
\indentrel{3}\begin{verbatim}
   (5)
             1  2   1  3    1  4    1   5    1   6
     1 + y + Ä y  + Ä y  + ÄÄ y  + ÄÄÄ y  + ÄÄÄ y
             2      6      24      120      720
   + 
     1   7     1    8      1    9      1     10      11
   ÄÄÄÄ y  + ÄÄÄÄÄ y  + ÄÄÄÄÄÄ y  + ÄÄÄÄÄÄÄ y   + O(y  )
   5040      40320      362880      3628800
     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty5}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty5}{ugxProblemSeriesFunctionsPagePatch5}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty5}{\showpaste}
\tab{5}\spadcommand{exp(y)\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch6}
\begin{paste}{ugxProblemSeriesFunctionsPageFull6}{ugxProblemSeriesFunctionsPageEmpty6}
\pastebutton{ugxProblemSeriesFunctionsPageFull6}{\hidepaste}
\tab{5}\spadcommand{tan(y**2)\free{y }}
\indentrel{3}\begin{verbatim}
         2   1  6    2  10      11
   (6)  y  + Ä y  + ÄÄ y   + O(y  )
             3      15
     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty6}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty6}{ugxProblemSeriesFunctionsPagePatch6}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty6}{\showpaste}
\tab{5}\spadcommand{tan(y**2)\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch7}
\begin{paste}{ugxProblemSeriesFunctionsPageFull7}{ugxProblemSeriesFunctionsPageEmpty7}
\pastebutton{ugxProblemSeriesFunctionsPageFull7}{\hidepaste}
\tab{5}\spadcommand{cos(y + y**5)\free{y }}
\indentrel{3}\begin{verbatim}
   (7)
         1  2    1  4   721  6    6721  8   1844641  10
     1 - Ä y  + ÄÄ y  - ÄÄÄ y  + ÄÄÄÄÄ y  - ÄÄÄÄÄÄÄ y
         2      24      720      40320      3628800
   + 
        11
     O(y  )
     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty7}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty7}{ugxProblemSeriesFunctionsPagePatch7}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty7}{\showpaste}
\tab{5}\spadcommand{cos(y + y**5)\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch8}
\begin{paste}{ugxProblemSeriesFunctionsPageFull8}{ugxProblemSeriesFunctionsPageEmpty8}
\pastebutton{ugxProblemSeriesFunctionsPageFull8}{\hidepaste}
\tab{5}\spadcommand{log(1 + sin(y))\free{y }}
\indentrel{3}\begin{verbatim}
   (8)
         1  2   1  3    1  4    1  5    1  6    61   7
     y - Ä y  + Ä y  - ÄÄ y  + ÄÄ y  - ÄÄ y  + ÄÄÄÄ y
         2      6      12      24      45      5040
   + 
        17   8    277   9     31   10      11
     - ÄÄÄÄ y  + ÄÄÄÄÄ y  - ÄÄÄÄÄ y   + O(y  )
       2520      72576      14175
     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty8}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty8}{ugxProblemSeriesFunctionsPagePatch8}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty8}{\showpaste}
\tab{5}\spadcommand{log(1 + sin(y))\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch9}
\begin{paste}{ugxProblemSeriesFunctionsPageFull9}{ugxProblemSeriesFunctionsPageEmpty9}
\pastebutton{ugxProblemSeriesFunctionsPageFull9}{\hidepaste}
\tab{5}\spadcommand{z : UTS(EXPR INT,z,0) := z\bound{z }}
\indentrel{3}\begin{verbatim}
   (9)  z
   Type: UnivariateTaylorSeries(Expression Integer,z,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty9}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty9}{ugxProblemSeriesFunctionsPagePatch9}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty9}{\showpaste}
\tab{5}\spadcommand{z : UTS(EXPR INT,z,0) := z\bound{z }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch10}
\begin{paste}{ugxProblemSeriesFunctionsPageFull10}{ugxProblemSeriesFunctionsPageEmpty10}
\pastebutton{ugxProblemSeriesFunctionsPageFull10}{\hidepaste}
\tab{5}\spadcommand{exp(2 + tan(z))\free{z }}
\indentrel{3}\begin{verbatim}
   (10)
                    2        2         2          2
       2     2    %e   2   %e   3   3%e   4   37%e   5
     %e  + %e z + ÄÄÄ z  + ÄÄÄ z  + ÄÄÄÄ z  + ÄÄÄÄÄ z
                   2        2         8        120
   + 
         2           2           2             2
     59%e   6   137%e   7   871%e   8   41641%e   9
     ÄÄÄÄÄ z  + ÄÄÄÄÄÄ z  + ÄÄÄÄÄÄ z  + ÄÄÄÄÄÄÄÄ z
      240         720        5760        362880
   + 
             2
     325249%e   10      11
     ÄÄÄÄÄÄÄÄÄ z   + O(z  )
      3628800
   Type: UnivariateTaylorSeries(Expression Integer,z,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty10}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty10}{ugxProblemSeriesFunctionsPagePatch10}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty10}{\showpaste}
\tab{5}\spadcommand{exp(2 + tan(z))\free{z }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch11}
\begin{paste}{ugxProblemSeriesFunctionsPageFull11}{ugxProblemSeriesFunctionsPageEmpty11}
\pastebutton{ugxProblemSeriesFunctionsPageFull11}{\hidepaste}
\tab{5}\spadcommand{w := taylor 'w\bound{w }}
\indentrel{3}\begin{verbatim}
   (11)  w
   Type: UnivariateTaylorSeries(Expression Integer,w,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty11}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty11}{ugxProblemSeriesFunctionsPagePatch11}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty11}{\showpaste}
\tab{5}\spadcommand{w := taylor 'w\bound{w }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPagePatch12}
\begin{paste}{ugxProblemSeriesFunctionsPageFull12}{ugxProblemSeriesFunctionsPageEmpty12}
\pastebutton{ugxProblemSeriesFunctionsPageFull12}{\hidepaste}
\tab{5}\spadcommand{exp(2 + tan(w))\free{w }}
\indentrel{3}\begin{verbatim}
   (12)
                    2        2         2          2
       2     2    %e   2   %e   3   3%e   4   37%e   5
     %e  + %e w + ÄÄÄ w  + ÄÄÄ w  + ÄÄÄÄ w  + ÄÄÄÄÄ w
                   2        2         8        120
   + 
         2           2           2             2
     59%e   6   137%e   7   871%e   8   41641%e   9
     ÄÄÄÄÄ w  + ÄÄÄÄÄÄ w  + ÄÄÄÄÄÄ w  + ÄÄÄÄÄÄÄÄ w
      240         720        5760        362880
   + 
             2
     325249%e   10      11
     ÄÄÄÄÄÄÄÄÄ w   + O(w  )
      3628800
   Type: UnivariateTaylorSeries(Expression Integer,w,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFunctionsPageEmpty12}
\begin{paste}{ugxProblemSeriesFunctionsPageEmpty12}{ugxProblemSeriesFunctionsPagePatch12}
\pastebutton{ugxProblemSeriesFunctionsPageEmpty12}{\showpaste}
\tab{5}\spadcommand{exp(2 + tan(w))\free{w }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch1}
\begin{paste}{ugProblemGeneticPageFull1}{ugProblemGeneticPageEmpty1}
\pastebutton{ugProblemGeneticPageFull1}{\hidepaste}
\tab{5}\spadcommand{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ]\bound{segregationRates }}
\indentrel{3}\begin{verbatim}
   (1)
    Ú    1   1    9¿  Ú    1        1¿  Ú        1    1¿
    ³1   Ä   Ä   Äij  ³0   Ä   0   Äij  ³0   0   Ä   Äij
    ³    2   2   20³  ³    2       20³  ³        2   20³
    ³              ³  ³              ³  ³              ³
    ³1        1    ³  ³1        9  1 ³  ³         9    ³
    ³Ä   0   ÄÄ  0 ³  ³Ä   1   ÄÄ  Ä ³  ³0   0   ÄÄ  0 ³
    ³2       20    ³  ³2       20  2 ³  ³        20    ³
   [³              ³, ³              ³, ³              ³,
    ³1    1        ³  ³     9        ³  ³1    9      1 ³
    ³Ä   ÄÄ  0   0 ³  ³0   ÄÄ  0   0 ³  ³Ä   ÄÄ  1   Ä ³
    ³2   20        ³  ³    20        ³  ³2   20      2 ³
    ³              ³  ³              ³  ³              ³
    ³ 9            ³  ³ 1  1         ³  ³ 1      1     ³
    ³ÄÄ  0   0   0 ³  ³ÄÄ  Ä   0   0 ³  ³ÄÄ  0   Ä   0 ³
    À20            Ù  À20  2         Ù  À20      2     Ù
    Ú             9¿
    ³0   0   0   Äij
    ³            20³
    ³              ³
    ³         1  1 ³
    ³0   0   ÄÄ  Ä ³
    ³        20  2 ³
    ³              ³]
    ³     1      1 ³
    ³0   ÄÄ  0   Ä ³
    ³    20      2 ³
    ³              ³
    ³ 9  1   1     ³
    ³ÄÄ  Ä   Ä   1 ³
    À20  2   2     Ù
            Type: List SquareMatrix(4,Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty1}
\begin{paste}{ugProblemGeneticPageEmpty1}{ugProblemGeneticPagePatch1}
\pastebutton{ugProblemGeneticPageEmpty1}{\showpaste}
\tab{5}\spadcommand{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ]\bound{segregationRates }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch2}
\begin{paste}{ugProblemGeneticPageFull2}{ugProblemGeneticPageEmpty2}
\pastebutton{ugProblemGeneticPageFull2}{\hidepaste}
\tab{5}\spadcommand{gametes := ['AB,'Ab,'aB,'ab]\bound{gametes }}
\indentrel{3}\begin{verbatim}
   (2)  [AB,Ab,aB,ab]
           Type: List OrderedVariableList [AB,Ab,aB,ab]
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty2}
\begin{paste}{ugProblemGeneticPageEmpty2}{ugProblemGeneticPagePatch2}
\pastebutton{ugProblemGeneticPageEmpty2}{\showpaste}
\tab{5}\spadcommand{gametes := ['AB,'Ab,'aB,'ab]\bound{gametes }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch3}
\begin{paste}{ugProblemGeneticPageFull3}{ugProblemGeneticPageEmpty3}
\pastebutton{ugProblemGeneticPageFull3}{\hidepaste}
\tab{5}\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A }\free{gametes segregationRates }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty3}
\begin{paste}{ugProblemGeneticPageEmpty3}{ugProblemGeneticPagePatch3}
\pastebutton{ugProblemGeneticPageEmpty3}{\showpaste}
\tab{5}\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A }\free{gametes segregationRates }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch4}
\begin{paste}{ugProblemGeneticPageFull4}{ugProblemGeneticPageEmpty4}
\pastebutton{ugProblemGeneticPageFull4}{\hidepaste}
\tab{5}\spadcommand{a := basis()$A; a.1*a.4}
\indentrel{3}\begin{verbatim}
         9       1       1       9
   (4)  ÄÄ ab + ÄÄ aB + ÄÄ Ab + ÄÄ AB
        20      20      20      20
Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty4}
\begin{paste}{ugProblemGeneticPageEmpty4}{ugProblemGeneticPagePatch4}
\pastebutton{ugProblemGeneticPageEmpty4}{\showpaste}
\tab{5}\spadcommand{a := basis()$A; a.1*a.4}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch5}
\begin{paste}{ugProblemGeneticPageFull5}{ugProblemGeneticPageEmpty5}
\pastebutton{ugProblemGeneticPageFull5}{\hidepaste}
\tab{5}\spadcommand{commutative?()$A\free{A }}
\indentrel{3}\begin{verbatim}
   algebra is commutative
   (5)  true
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty5}
\begin{paste}{ugProblemGeneticPageEmpty5}{ugProblemGeneticPagePatch5}
\pastebutton{ugProblemGeneticPageEmpty5}{\showpaste}
\tab{5}\spadcommand{commutative?()$A\free{A }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch6}
\begin{paste}{ugProblemGeneticPageFull6}{ugProblemGeneticPageEmpty6}
\pastebutton{ugProblemGeneticPageFull6}{\hidepaste}
\tab{5}\spadcommand{associative?()$A\free{A }}
\indentrel{3}\begin{verbatim}
   algebra is not associative
   (6)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty6}
\begin{paste}{ugProblemGeneticPageEmpty6}{ugProblemGeneticPagePatch6}
\pastebutton{ugProblemGeneticPageEmpty6}{\showpaste}
\tab{5}\spadcommand{associative?()$A\free{A }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch7}
\begin{paste}{ugProblemGeneticPageFull7}{ugProblemGeneticPageEmpty7}
\pastebutton{ugProblemGeneticPageFull7}{\hidepaste}
\tab{5}\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]\bound{x }\free{gametes segregationRates }}
\indentrel{3}\begin{verbatim}
   (7)  0.4ab + 0.1aB + 0.2Ab + 0.3AB
Type: AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty7}
\begin{paste}{ugProblemGeneticPageEmpty7}{ugProblemGeneticPagePatch7}
\pastebutton{ugProblemGeneticPageEmpty7}{\showpaste}
\tab{5}\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]\bound{x }\free{gametes segregationRates }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch8}
\begin{paste}{ugProblemGeneticPageFull8}{ugProblemGeneticPageEmpty8}
\pastebutton{ugProblemGeneticPageFull8}{\hidepaste}
\tab{5}\spadcommand{plenaryPower(x,5)\free{x }}
\indentrel{3}\begin{verbatim}
   (8)  0.36561ab + 0.13439aB + 0.23439Ab + 0.26561AB
Type: AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty8}
\begin{paste}{ugProblemGeneticPageEmpty8}{ugProblemGeneticPagePatch8}
\pastebutton{ugProblemGeneticPageEmpty8}{\showpaste}
\tab{5}\spadcommand{plenaryPower(x,5)\free{x }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch9}
\begin{paste}{ugProblemGeneticPageFull9}{ugProblemGeneticPageEmpty9}
\pastebutton{ugProblemGeneticPageFull9}{\hidepaste}
\tab{5}\spadcommand{q := leftRankPolynomial()$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q }\free{gametes segregationRates }}
\indentrel{3}\begin{verbatim}
   (9)
      3      29       29       29       29      2
     Y  + (- ÄÄ %x4 - ÄÄ %x3 - ÄÄ %x2 - ÄÄ %x1)Y
             20       20       20       20
   + 
          9    2     9        9        9            9    2
         ÄÄ %x4  + (ÄÄ %x3 + ÄÄ %x2 + ÄÄ %x1)%x4 + ÄÄ %x3
         20         10       10       10           20
       + 
           9        9            9    2    9
         (ÄÄ %x2 + ÄÄ %x1)%x3 + ÄÄ %x2  + ÄÄ %x1 %x2
          10       10           20        10
       + 
          9    2
         ÄÄ %x1
         20
    *
       Y
Type: UnivariatePolynomial(Y,Polynomial Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty9}
\begin{paste}{ugProblemGeneticPageEmpty9}{ugProblemGeneticPagePatch9}
\pastebutton{ugProblemGeneticPageEmpty9}{\showpaste}
\tab{5}\spadcommand{q := leftRankPolynomial()$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q }\free{gametes segregationRates }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch10}
\begin{paste}{ugProblemGeneticPageFull10}{ugProblemGeneticPageEmpty10}
\pastebutton{ugProblemGeneticPageFull10}{\hidepaste}
\tab{5}\spadcommand{factor(q :: POLY FRAC INT)\free{q }}
\indentrel{3}\begin{verbatim}
   (10)
     (Y - %x4 - %x3 - %x2 - %x1)
  *
           9        9        9        9
     (Y - ÄÄ %x4 - ÄÄ %x3 - ÄÄ %x2 - ÄÄ %x1)Y
          20       20       20       20
             Type: Factored Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty10}
\begin{paste}{ugProblemGeneticPageEmpty10}{ugProblemGeneticPagePatch10}
\pastebutton{ugProblemGeneticPageEmpty10}{\showpaste}
\tab{5}\spadcommand{factor(q :: POLY FRAC INT)\free{q }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch11}
\begin{paste}{ugProblemGeneticPageFull11}{ugProblemGeneticPageEmpty11}
\pastebutton{ugProblemGeneticPageFull11}{\hidepaste}
\tab{5}\spadcommand{cI := conditionsForIdempotents()$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI }\free{gametes segregationRates }}
\indentrel{3}\begin{verbatim}
   (11)
     9             1                              2
   [ÄÄ %x1 %x4 + (ÄÄ %x2 + %x1)%x3 + %x1 %x2 + %x1  - %x1,
    10            10
            1            9              2
    (%x2 + ÄÄ %x1)%x4 + ÄÄ %x2 %x3 + %x2  + (%x1 - 1)%x2,
           10           10
            1              2     9
    (%x3 + ÄÄ %x1)%x4 + %x3  + (ÄÄ %x2 + %x1 - 1)%x3,
           10                   10
       2                 9                1
    %x4  + (%x3 + %x2 + ÄÄ %x1 - 1)%x4 + ÄÄ %x2 %x3]
                        10               10
                 Type: List Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty11}
\begin{paste}{ugProblemGeneticPageEmpty11}{ugProblemGeneticPagePatch11}
\pastebutton{ugProblemGeneticPageEmpty11}{\showpaste}
\tab{5}\spadcommand{cI := conditionsForIdempotents()$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI }\free{gametes segregationRates }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch12}
\begin{paste}{ugProblemGeneticPageFull12}{ugProblemGeneticPageEmpty12}
\pastebutton{ugProblemGeneticPageFull12}{\hidepaste}
\tab{5}\spadcommand{gbs:= groebnerFactorize cI; gbs.1\free{cI }\bound{gbs }}
\indentrel{3}\begin{verbatim}
   (12)
   [%x4 + %x3 + %x2 + %x1 - 1,
                                  2
    (%x2 + %x1)%x3 + %x1 %x2 + %x1  - %x1]
                 Type: List Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty12}
\begin{paste}{ugProblemGeneticPageEmpty12}{ugProblemGeneticPagePatch12}
\pastebutton{ugProblemGeneticPageEmpty12}{\showpaste}
\tab{5}\spadcommand{gbs:= groebnerFactorize cI; gbs.1\free{cI }\bound{gbs }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch13}
\begin{paste}{ugProblemGeneticPageFull13}{ugProblemGeneticPageEmpty13}
\pastebutton{ugProblemGeneticPageFull13}{\hidepaste}
\tab{5}\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol }\free{gbs }}
\indentrel{3}\begin{verbatim}
                2      2       1       1
   (13)  [[%x4= Ä,%x3= Ä,%x2= ÄÄ,%x1= ÄÄ]]
                5      5      10      10
   Type: List List Equation Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty13}
\begin{paste}{ugProblemGeneticPageEmpty13}{ugProblemGeneticPagePatch13}
\pastebutton{ugProblemGeneticPageEmpty13}{\showpaste}
\tab{5}\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol }\free{gbs }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch14}
\begin{paste}{ugProblemGeneticPageFull14}{ugProblemGeneticPageEmpty14}
\pastebutton{ugProblemGeneticPageFull14}{\hidepaste}
\tab{5}\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e }\free{A sol }}
\indentrel{3}\begin{verbatim}
         2      2       1       1
   (14)  Ä ab + Ä aB + ÄÄ Ab + ÄÄ AB
         5      5      10      10
Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty14}
\begin{paste}{ugProblemGeneticPageEmpty14}{ugProblemGeneticPagePatch14}
\pastebutton{ugProblemGeneticPageEmpty14}{\showpaste}
\tab{5}\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e }\free{A sol }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPagePatch15}
\begin{paste}{ugProblemGeneticPageFull15}{ugProblemGeneticPageEmpty15}
\pastebutton{ugProblemGeneticPageFull15}{\hidepaste}
\tab{5}\spadcommand{e*e-e\free{e }}
\indentrel{3}\begin{verbatim}
   (15)  0
Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGeneticPageEmpty15}
\begin{paste}{ugProblemGeneticPageEmpty15}{ugProblemGeneticPagePatch15}
\pastebutton{ugProblemGeneticPageEmpty15}{\showpaste}
\tab{5}\spadcommand{e*e-e\free{e }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPagePatch1}
\begin{paste}{ugxProblemOnePolPageFull1}{ugxProblemOnePolPageEmpty1}
\pastebutton{ugxProblemOnePolPageFull1}{\hidepaste}
\tab{5}\spadcommand{solve(x**3 = 8,x)}
\indentrel{3}\begin{verbatim}
               2
   (1)  [x= 2,x  + 2x + 4= 0]
        Type: List Equation Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPageEmpty1}
\begin{paste}{ugxProblemOnePolPageEmpty1}{ugxProblemOnePolPagePatch1}
\pastebutton{ugxProblemOnePolPageEmpty1}{\showpaste}
\tab{5}\spadcommand{solve(x**3 = 8,x)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPagePatch2}
\begin{paste}{ugxProblemOnePolPageFull2}{ugxProblemOnePolPageEmpty2}
\pastebutton{ugxProblemOnePolPageFull2}{\hidepaste}
\tab{5}\spadcommand{radicalSolve(x**3 = 8,x)}
\indentrel{3}\begin{verbatim}
               ÚÄÄÄ¿         ÚÄÄÄ¿
   (2)  [x= - \³- 3  - 1,x= \³- 3  - 1,x= 2]
                 Type: List Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPageEmpty2}
\begin{paste}{ugxProblemOnePolPageEmpty2}{ugxProblemOnePolPagePatch2}
\pastebutton{ugxProblemOnePolPageEmpty2}{\showpaste}
\tab{5}\spadcommand{radicalSolve(x**3 = 8,x)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPagePatch3}
\begin{paste}{ugxProblemOnePolPageFull3}{ugxProblemOnePolPageEmpty3}
\pastebutton{ugxProblemOnePolPageFull3}{\hidepaste}
\tab{5}\spadcommand{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)}
\indentrel{3}\begin{verbatim}
   (3)  [x= 3.6180114746 09375,x= 1.3819885253 90625]
                   Type: List Equation Polynomial Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPageEmpty3}
\begin{paste}{ugxProblemOnePolPageEmpty3}{ugxProblemOnePolPagePatch3}
\pastebutton{ugxProblemOnePolPageEmpty3}{\showpaste}
\tab{5}\spadcommand{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPagePatch4}
\begin{paste}{ugxProblemOnePolPageFull4}{ugxProblemOnePolPageEmpty4}
\pastebutton{ugxProblemOnePolPageFull4}{\hidepaste}
\tab{5}\spadcommand{solve(x**3-2,1/1000)}
\indentrel{3}\begin{verbatim}
            2581
   (4)  [x= ÄÄÄÄ]
            2048
        Type: List Equation Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPageEmpty4}
\begin{paste}{ugxProblemOnePolPageEmpty4}{ugxProblemOnePolPagePatch4}
\pastebutton{ugxProblemOnePolPageEmpty4}{\showpaste}
\tab{5}\spadcommand{solve(x**3-2,1/1000)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPagePatch5}
\begin{paste}{ugxProblemOnePolPageFull5}{ugxProblemOnePolPageEmpty5}
\pastebutton{ugxProblemOnePolPageFull5}{\hidepaste}
\tab{5}\spadcommand{complexSolve(x**3-2,.0001)}
\indentrel{3}\begin{verbatim}
   (5)
   [x= 1.2599182128 90625,
    x= - 0.6298943279 5395613131 - 1.0910949707 03125 %i,
    x= - 0.6298943279 5395613131 + 1.0910949707 03125 %i]
           Type: List Equation Polynomial Complex Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPageEmpty5}
\begin{paste}{ugxProblemOnePolPageEmpty5}{ugxProblemOnePolPagePatch5}
\pastebutton{ugxProblemOnePolPageEmpty5}{\showpaste}
\tab{5}\spadcommand{complexSolve(x**3-2,.0001)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPagePatch6}
\begin{paste}{ugxProblemOnePolPageFull6}{ugxProblemOnePolPageEmpty6}
\pastebutton{ugxProblemOnePolPageFull6}{\hidepaste}
\tab{5}\spadcommand{complexSolve(x**2-2*\%i+1,1/100)}
\indentrel{3}\begin{verbatim}
              13028925   325       13028925   325
   (6)  [x= - ÄÄÄÄÄÄÄÄ - ÄÄÄ %i,x= ÄÄÄÄÄÄÄÄ + ÄÄÄ %i]
              16777216   256       16777216   256
Type: List Equation Polynomial Complex Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPageEmpty6}
\begin{paste}{ugxProblemOnePolPageEmpty6}{ugxProblemOnePolPagePatch6}
\pastebutton{ugxProblemOnePolPageEmpty6}{\showpaste}
\tab{5}\spadcommand{complexSolve(x**2-2*\%i+1,1/100)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPagePatch7}
\begin{paste}{ugxProblemOnePolPageFull7}{ugxProblemOnePolPageEmpty7}
\pastebutton{ugxProblemOnePolPageFull7}{\hidepaste}
\tab{5}\spadcommand{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)}
\indentrel{3}\begin{verbatim}
               ÚÄÄÄ¿         ÚÄÄÄ¿
            - \³- 3  - 1    \³- 3  - 1
   (7)  [x= ÄÄÄÄÄÄÄÄÄÄÄÄ,x= ÄÄÄÄÄÄÄÄÄÄ]
                  2              2
                 Type: List Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemOnePolPageEmpty7}
\begin{paste}{ugxProblemOnePolPageEmpty7}{ugxProblemOnePolPagePatch7}
\pastebutton{ugxProblemOnePolPageEmpty7}{\showpaste}
\tab{5}\spadcommand{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch1}
\begin{paste}{ugProblemNumericPageFull1}{ugProblemNumericPageEmpty1}
\pastebutton{ugProblemNumericPageFull1}{\hidepaste}
\tab{5}\spadcommand{exp(3.1)}
\indentrel{3}\begin{verbatim}
   (1)  22.1979512814 41633405
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty1}
\begin{paste}{ugProblemNumericPageEmpty1}{ugProblemNumericPagePatch1}
\pastebutton{ugProblemNumericPageEmpty1}{\showpaste}
\tab{5}\spadcommand{exp(3.1)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch2}
\begin{paste}{ugProblemNumericPageFull2}{ugProblemNumericPageEmpty2}
\pastebutton{ugProblemNumericPageFull2}{\hidepaste}
\tab{5}\spadcommand{exp(3.1 + 4.5 * \%i)}
\indentrel{3}\begin{verbatim}
   (2)
   - 4.6792348860 969899118 - 21.6991659280 71731864 %i
                                    Type: Complex Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty2}
\begin{paste}{ugProblemNumericPageEmpty2}{ugProblemNumericPagePatch2}
\pastebutton{ugProblemNumericPageEmpty2}{\showpaste}
\tab{5}\spadcommand{exp(3.1 + 4.5 * \%i)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch3}
\begin{paste}{ugProblemNumericPageFull3}{ugProblemNumericPageEmpty3}
\pastebutton{ugProblemNumericPageFull3}{\hidepaste}
\tab{5}\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)}
\indentrel{3}\begin{verbatim}
   (3)  - 4.6792348860969906 - 21.699165928071732%i
                              Type: Complex DoubleFloat
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty3}
\begin{paste}{ugProblemNumericPageEmpty3}{ugProblemNumericPagePatch3}
\pastebutton{ugProblemNumericPageEmpty3}{\showpaste}
\tab{5}\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch4}
\begin{paste}{ugProblemNumericPageFull4}{ugProblemNumericPageEmpty4}
\pastebutton{ugProblemNumericPageFull4}{\hidepaste}
\tab{5}\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)}
\indentrel{3}\begin{verbatim}
   (4)  - 4.6792348860969906 - 21.699165928071732%i
                              Type: Complex DoubleFloat
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty4}
\begin{paste}{ugProblemNumericPageEmpty4}{ugProblemNumericPagePatch4}
\pastebutton{ugProblemNumericPageEmpty4}{\showpaste}
\tab{5}\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch5}
\begin{paste}{ugProblemNumericPageFull5}{ugProblemNumericPageEmpty5}
\pastebutton{ugProblemNumericPageFull5}{\hidepaste}
\tab{5}\spadcommand{Gamma(0.5)**2}
\indentrel{3}\begin{verbatim}
   (5)  3.14159265358979
                                      Type: DoubleFloat
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty5}
\begin{paste}{ugProblemNumericPageEmpty5}{ugProblemNumericPagePatch5}
\pastebutton{ugProblemNumericPageEmpty5}{\showpaste}
\tab{5}\spadcommand{Gamma(0.5)**2}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch6}
\begin{paste}{ugProblemNumericPageFull6}{ugProblemNumericPageEmpty6}
\pastebutton{ugProblemNumericPageFull6}{\hidepaste}
\tab{5}\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)}
\indentrel{3}\begin{verbatim}
   (6)  2.489482417547372 - 2.3658460381468371%i
                              Type: Complex DoubleFloat
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty6}
\begin{paste}{ugProblemNumericPageEmpty6}{ugProblemNumericPagePatch6}
\pastebutton{ugProblemNumericPageEmpty6}{\showpaste}
\tab{5}\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch7}
\begin{paste}{ugProblemNumericPageFull7}{ugProblemNumericPageEmpty7}
\pastebutton{ugProblemNumericPageFull7}{\hidepaste}
\tab{5}\spadcommand{[chebyshevT(i, z) for i in 0..5]}
\indentrel{3}\begin{verbatim}
   (7)
          2       3        4     2        5      3
   [1,z,2z  - 1,4z  - 3z,8z  - 8z  + 1,16z  - 20z  + 5z]
                          Type: List Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty7}
\begin{paste}{ugProblemNumericPageEmpty7}{ugProblemNumericPagePatch7}
\pastebutton{ugProblemNumericPageEmpty7}{\showpaste}
\tab{5}\spadcommand{[chebyshevT(i, z) for i in 0..5]}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch8}
\begin{paste}{ugProblemNumericPageFull8}{ugProblemNumericPageEmpty8}
\pastebutton{ugProblemNumericPageFull8}{\hidepaste}
\tab{5}\spadcommand{chebyshevT(3, 5.0 + 6.0*\%i)}
\indentrel{3}\begin{verbatim}
   (8)  - 1675.0 + 918.0 %i
                                    Type: Complex Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty8}
\begin{paste}{ugProblemNumericPageEmpty8}{ugProblemNumericPagePatch8}
\pastebutton{ugProblemNumericPageEmpty8}{\showpaste}
\tab{5}\spadcommand{chebyshevT(3, 5.0 + 6.0*\%i)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch9}
\begin{paste}{ugProblemNumericPageFull9}{ugProblemNumericPageEmpty9}
\pastebutton{ugProblemNumericPageFull9}{\hidepaste}
\tab{5}\spadcommand{chebyshevT(3, 5.0::DoubleFloat)}
\indentrel{3}\begin{verbatim}
   (9)  485.0
                                      Type: DoubleFloat
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty9}
\begin{paste}{ugProblemNumericPageEmpty9}{ugProblemNumericPagePatch9}
\pastebutton{ugProblemNumericPageEmpty9}{\showpaste}
\tab{5}\spadcommand{chebyshevT(3, 5.0::DoubleFloat)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch10}
\begin{paste}{ugProblemNumericPageFull10}{ugProblemNumericPageEmpty10}
\pastebutton{ugProblemNumericPageFull10}{\hidepaste}
\tab{5}\spadcommand{[chebyshevU(i, z) for i in 0..5]}
\indentrel{3}\begin{verbatim}
   (10)
             2        3          4      2
   [1, 2z, 4z  - 1, 8z  - 4z, 16z  - 12z  + 1,
       5      3
    32z  - 32z  + 6z]
                          Type: List Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty10}
\begin{paste}{ugProblemNumericPageEmpty10}{ugProblemNumericPagePatch10}
\pastebutton{ugProblemNumericPageEmpty10}{\showpaste}
\tab{5}\spadcommand{[chebyshevU(i, z) for i in 0..5]}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch11}
\begin{paste}{ugProblemNumericPageFull11}{ugProblemNumericPageEmpty11}
\pastebutton{ugProblemNumericPageFull11}{\hidepaste}
\tab{5}\spadcommand{chebyshevU(3, 0.2)}
\indentrel{3}\begin{verbatim}
   (11)  - 0.736
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty11}
\begin{paste}{ugProblemNumericPageEmpty11}{ugProblemNumericPagePatch11}
\pastebutton{ugProblemNumericPageEmpty11}{\showpaste}
\tab{5}\spadcommand{chebyshevU(3, 0.2)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch12}
\begin{paste}{ugProblemNumericPageFull12}{ugProblemNumericPageEmpty12}
\pastebutton{ugProblemNumericPageFull12}{\hidepaste}
\tab{5}\spadcommand{[hermiteH(i, z) for i in 0..5]}
\indentrel{3}\begin{verbatim}
   (12)
             2        3           4      2
   [1, 2z, 4z  - 2, 8z  - 12z, 16z  - 48z  + 12,
       5       3
    32z  - 160z  + 120z]
                          Type: List Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty12}
\begin{paste}{ugProblemNumericPageEmpty12}{ugProblemNumericPagePatch12}
\pastebutton{ugProblemNumericPageEmpty12}{\showpaste}
\tab{5}\spadcommand{[hermiteH(i, z) for i in 0..5]}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch13}
\begin{paste}{ugProblemNumericPageFull13}{ugProblemNumericPageEmpty13}
\pastebutton{ugProblemNumericPageFull13}{\hidepaste}
\tab{5}\spadcommand{hermiteH(100, 1.0)}
\indentrel{3}\begin{verbatim}
   (13)  - 0.1448706729 337934088 E 93
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty13}
\begin{paste}{ugProblemNumericPageEmpty13}{ugProblemNumericPagePatch13}
\pastebutton{ugProblemNumericPageEmpty13}{\showpaste}
\tab{5}\spadcommand{hermiteH(100, 1.0)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch14}
\begin{paste}{ugProblemNumericPageFull14}{ugProblemNumericPageEmpty14}
\pastebutton{ugProblemNumericPageFull14}{\hidepaste}
\tab{5}\spadcommand{[laguerreL(i, z) for i in 0..4]}
\indentrel{3}\begin{verbatim}
   (14)
                 2              3     2
   [1, - z + 1, z  - 4z + 2, - z  + 9z  - 18z + 6,
     4      3      2
    z  - 16z  + 72z  - 96z + 24]
                          Type: List Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty14}
\begin{paste}{ugProblemNumericPageEmpty14}{ugProblemNumericPagePatch14}
\pastebutton{ugProblemNumericPageEmpty14}{\showpaste}
\tab{5}\spadcommand{[laguerreL(i, z) for i in 0..4]}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch15}
\begin{paste}{ugProblemNumericPageFull15}{ugProblemNumericPageEmpty15}
\pastebutton{ugProblemNumericPageFull15}{\hidepaste}
\tab{5}\spadcommand{laguerreL(4, 1.2)}
\indentrel{3}\begin{verbatim}
   (15)  - 13.0944
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty15}
\begin{paste}{ugProblemNumericPageEmpty15}{ugProblemNumericPagePatch15}
\pastebutton{ugProblemNumericPageEmpty15}{\showpaste}
\tab{5}\spadcommand{laguerreL(4, 1.2)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch16}
\begin{paste}{ugProblemNumericPageFull16}{ugProblemNumericPageEmpty16}
\pastebutton{ugProblemNumericPageFull16}{\hidepaste}
\tab{5}\spadcommand{[laguerreL(j, 3, z) for j in 0..4]}
\indentrel{3}\begin{verbatim}
   (16)
       3     2               2
   [- z  + 9z  - 18z + 6,- 3z  + 18z - 18,- 6z + 18,- 6,0]
                          Type: List Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty16}
\begin{paste}{ugProblemNumericPageEmpty16}{ugProblemNumericPagePatch16}
\pastebutton{ugProblemNumericPageEmpty16}{\showpaste}
\tab{5}\spadcommand{[laguerreL(j, 3, z) for j in 0..4]}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch17}
\begin{paste}{ugProblemNumericPageFull17}{ugProblemNumericPageEmpty17}
\pastebutton{ugProblemNumericPageFull17}{\hidepaste}
\tab{5}\spadcommand{laguerreL(1, 3, 2.1)}
\indentrel{3}\begin{verbatim}
   (17)  6.57
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty17}
\begin{paste}{ugProblemNumericPageEmpty17}{ugProblemNumericPagePatch17}
\pastebutton{ugProblemNumericPageEmpty17}{\showpaste}
\tab{5}\spadcommand{laguerreL(1, 3, 2.1)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch18}
\begin{paste}{ugProblemNumericPageFull18}{ugProblemNumericPageEmpty18}
\pastebutton{ugProblemNumericPageFull18}{\hidepaste}
\tab{5}\spadcommand{[legendreP(i,z) for i in 0..5]}
\indentrel{3}\begin{verbatim}
   (18)
          3  2   1  5  3   3    35  4   15  2   3
   [1, z, Ä z  - Ä, Ä z  - Ä z, ÄÄ z  - ÄÄ z  + Ä,
          2      2  2      2     8       4      8
    63  5   35  3   15
    ÄÄ z  - ÄÄ z  + ÄÄ z]
     8       4       8
                 Type: List Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty18}
\begin{paste}{ugProblemNumericPageEmpty18}{ugProblemNumericPagePatch18}
\pastebutton{ugProblemNumericPageEmpty18}{\showpaste}
\tab{5}\spadcommand{[legendreP(i,z) for i in 0..5]}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch19}
\begin{paste}{ugProblemNumericPageFull19}{ugProblemNumericPageEmpty19}
\pastebutton{ugProblemNumericPageFull19}{\hidepaste}
\tab{5}\spadcommand{legendreP(3, 3.0*\%i)}
\indentrel{3}\begin{verbatim}
   (19)  - 72.0 %i
                                    Type: Complex Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty19}
\begin{paste}{ugProblemNumericPageEmpty19}{ugProblemNumericPagePatch19}
\pastebutton{ugProblemNumericPageEmpty19}{\showpaste}
\tab{5}\spadcommand{legendreP(3, 3.0*\%i)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch20}
\begin{paste}{ugProblemNumericPageFull20}{ugProblemNumericPageEmpty20}
\pastebutton{ugProblemNumericPageFull20}{\hidepaste}
\tab{5}\spadcommand{bernoulliB(3, z)}
\indentrel{3}\begin{verbatim}
          3   3  2   1
   (20)  z  - Ä z  + Ä z
              2      2
                      Type: Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty20}
\begin{paste}{ugProblemNumericPageEmpty20}{ugProblemNumericPagePatch20}
\pastebutton{ugProblemNumericPageEmpty20}{\showpaste}
\tab{5}\spadcommand{bernoulliB(3, z)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch21}
\begin{paste}{ugProblemNumericPageFull21}{ugProblemNumericPageEmpty21}
\pastebutton{ugProblemNumericPageFull21}{\hidepaste}
\tab{5}\spadcommand{bernoulliB(3, 0.7 + 0.4 * \%i)}
\indentrel{3}\begin{verbatim}
   (21)  - 0.138 - 0.116 %i
                                    Type: Complex Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty21}
\begin{paste}{ugProblemNumericPageEmpty21}{ugProblemNumericPagePatch21}
\pastebutton{ugProblemNumericPageEmpty21}{\showpaste}
\tab{5}\spadcommand{bernoulliB(3, 0.7 + 0.4 * \%i)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch22}
\begin{paste}{ugProblemNumericPageFull22}{ugProblemNumericPageEmpty22}
\pastebutton{ugProblemNumericPageFull22}{\hidepaste}
\tab{5}\spadcommand{eulerE(3, z)}
\indentrel{3}\begin{verbatim}
          3   3  2   1
   (22)  z  - Ä z  + Ä
              2      4
                      Type: Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty22}
\begin{paste}{ugProblemNumericPageEmpty22}{ugProblemNumericPagePatch22}
\pastebutton{ugProblemNumericPageEmpty22}{\showpaste}
\tab{5}\spadcommand{eulerE(3, z)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch23}
\begin{paste}{ugProblemNumericPageFull23}{ugProblemNumericPageEmpty23}
\pastebutton{ugProblemNumericPageFull23}{\hidepaste}
\tab{5}\spadcommand{eulerE(3, 0.7 + 0.4 * \%i)}
\indentrel{3}\begin{verbatim}
   (23)  - 0.238 - 0.316 %i
                                    Type: Complex Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty23}
\begin{paste}{ugProblemNumericPageEmpty23}{ugProblemNumericPagePatch23}
\pastebutton{ugProblemNumericPageEmpty23}{\showpaste}
\tab{5}\spadcommand{eulerE(3, 0.7 + 0.4 * \%i)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch24}
\begin{paste}{ugProblemNumericPageFull24}{ugProblemNumericPageEmpty24}
\pastebutton{ugProblemNumericPageFull24}{\hidepaste}
\tab{5}\spadcommand{cyclotomic(3, z)}
\indentrel{3}\begin{verbatim}
          2
   (24)  z  + z + 1
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty24}
\begin{paste}{ugProblemNumericPageEmpty24}{ugProblemNumericPagePatch24}
\pastebutton{ugProblemNumericPageEmpty24}{\showpaste}
\tab{5}\spadcommand{cyclotomic(3, z)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch25}
\begin{paste}{ugProblemNumericPageFull25}{ugProblemNumericPageEmpty25}
\pastebutton{ugProblemNumericPageFull25}{\hidepaste}
\tab{5}\spadcommand{cyclotomic(3, (-1.0 + 0.0 * \%i)**(2/3))}
\indentrel{3}\begin{verbatim}
   (25)  0.0
                                    Type: Complex Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty25}
\begin{paste}{ugProblemNumericPageEmpty25}{ugProblemNumericPagePatch25}
\pastebutton{ugProblemNumericPageEmpty25}{\showpaste}
\tab{5}\spadcommand{cyclotomic(3, (-1.0 + 0.0 * \%i)**(2/3))}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch26}
\begin{paste}{ugProblemNumericPageFull26}{ugProblemNumericPageEmpty26}
\pastebutton{ugProblemNumericPageFull26}{\hidepaste}
\tab{5}\spadgraph{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +-> imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")}
\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage26.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage26}}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty26}
\begin{paste}{ugProblemNumericPageEmpty26}{ugProblemNumericPagePatch26}
\pastebutton{ugProblemNumericPageEmpty26}{\showpaste}
\tab{5}\spadgraph{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +-> imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch27}
\begin{paste}{ugProblemNumericPageFull27}{ugProblemNumericPageEmpty27}
\pastebutton{ugProblemNumericPageFull27}{\hidepaste}
\tab{5}\spadgraph{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp}
\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage27.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage27}}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty27}
\begin{paste}{ugProblemNumericPageEmpty27}{ugProblemNumericPagePatch27}
\pastebutton{ugProblemNumericPageEmpty27}{\showpaste}
\tab{5}\spadgraph{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch28}
\begin{paste}{ugProblemNumericPageFull28}{ugProblemNumericPageEmpty28}
\pastebutton{ugProblemNumericPageFull28}{\hidepaste}
\tab{5}\spadgraph{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)}
\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage28.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage28}}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty28}
\begin{paste}{ugProblemNumericPageEmpty28}{ugProblemNumericPagePatch28}
\pastebutton{ugProblemNumericPageEmpty28}{\showpaste}
\tab{5}\spadgraph{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch29}
\begin{paste}{ugProblemNumericPageFull29}{ugProblemNumericPageEmpty29}
\pastebutton{ugProblemNumericPageFull29}{\hidepaste}
\tab{5}\spadgraph{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)}
\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage29.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage29}}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty29}
\begin{paste}{ugProblemNumericPageEmpty29}{ugProblemNumericPagePatch29}
\pastebutton{ugProblemNumericPageEmpty29}{\showpaste}
\tab{5}\spadgraph{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch30}
\begin{paste}{ugProblemNumericPageFull30}{ugProblemNumericPageEmpty30}
\pastebutton{ugProblemNumericPageFull30}{\hidepaste}
\tab{5}\spadgraph{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)}
\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage30}}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty30}
\begin{paste}{ugProblemNumericPageEmpty30}{ugProblemNumericPagePatch30}
\pastebutton{ugProblemNumericPageEmpty30}{\showpaste}
\tab{5}\spadgraph{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch31}
\begin{paste}{ugProblemNumericPageFull31}{ugProblemNumericPageEmpty31}
\pastebutton{ugProblemNumericPageFull31}{\hidepaste}
\tab{5}\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])}
\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage31.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage31}}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty31}
\begin{paste}{ugProblemNumericPageEmpty31}{ugProblemNumericPagePatch31}
\pastebutton{ugProblemNumericPageEmpty31}{\showpaste}
\tab{5}\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch32}
\begin{paste}{ugProblemNumericPageFull32}{ugProblemNumericPageEmpty32}
\pastebutton{ugProblemNumericPageFull32}{\hidepaste}
\tab{5}\spadgraph{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")}
\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage32.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage32}}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty32}
\begin{paste}{ugProblemNumericPageEmpty32}{ugProblemNumericPagePatch32}
\pastebutton{ugProblemNumericPageEmpty32}{\showpaste}
\tab{5}\spadgraph{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")}
\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPagePatch33}
\begin{paste}{ugProblemNumericPageFull33}{ugProblemNumericPageEmpty33}
\pastebutton{ugProblemNumericPageFull33}{\hidepaste}
\tab{5}\spadcommand{all}
\indentrel{3}\begin{verbatim}
   (33)  all
                                     Type: Variable all
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemNumericPageEmpty33}
\begin{paste}{ugProblemNumericPageEmpty33}{ugProblemNumericPagePatch33}
\pastebutton{ugProblemNumericPageEmpty33}{\showpaste}
\tab{5}\spadcommand{all}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPagePatch1}
\begin{paste}{ugxProblemLinSysPageFull1}{ugxProblemLinSysPageEmpty1}
\pastebutton{ugxProblemLinSysPageFull1}{\hidepaste}
\tab{5}\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])}
\indentrel{3}\begin{verbatim}
   (1)  [[x= - 1,y= 2,z= 7]]
   Type: List List Equation Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPageEmpty1}
\begin{paste}{ugxProblemLinSysPageEmpty1}{ugxProblemLinSysPagePatch1}
\pastebutton{ugxProblemLinSysPageEmpty1}{\showpaste}
\tab{5}\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPagePatch2}
\begin{paste}{ugxProblemLinSysPageFull2}{ugxProblemLinSysPageEmpty2}
\pastebutton{ugxProblemLinSysPageFull2}{\hidepaste}
\tab{5}\spadcommand{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])}
\indentrel{3}\begin{verbatim}
   (2)  [[x= %Y - 2,y= - 2%Y + 2,z= %Y]]
   Type: List List Equation Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPageEmpty2}
\begin{paste}{ugxProblemLinSysPageEmpty2}{ugxProblemLinSysPagePatch2}
\pastebutton{ugxProblemLinSysPageEmpty2}{\showpaste}
\tab{5}\spadcommand{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPagePatch3}
\begin{paste}{ugxProblemLinSysPageFull3}{ugxProblemLinSysPageEmpty3}
\pastebutton{ugxProblemLinSysPageFull3}{\hidepaste}
\tab{5}\spadcommand{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])}
\indentrel{3}\begin{verbatim}
   (3)  [particular= [- 1,2,7],basis= [[0,0,0]]]
Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPageEmpty3}
\begin{paste}{ugxProblemLinSysPageEmpty3}{ugxProblemLinSysPagePatch3}
\pastebutton{ugxProblemLinSysPageEmpty3}{\showpaste}
\tab{5}\spadcommand{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPagePatch4}
\begin{paste}{ugxProblemLinSysPageFull4}{ugxProblemLinSysPageEmpty4}
\pastebutton{ugxProblemLinSysPageFull4}{\hidepaste}
\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])}
\indentrel{3}\begin{verbatim}
   (4)  [particular= [- 2,2,0],basis= [[1,- 2,1]]]
Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPageEmpty4}
\begin{paste}{ugxProblemLinSysPageEmpty4}{ugxProblemLinSysPagePatch4}
\pastebutton{ugxProblemLinSysPageEmpty4}{\showpaste}
\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPagePatch5}
\begin{paste}{ugxProblemLinSysPageFull5}{ugxProblemLinSysPageEmpty5}
\pastebutton{ugxProblemLinSysPageFull5}{\hidepaste}
\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])}
\indentrel{3}\begin{verbatim}
   (5)  [particular= "failed",basis= [[1,- 2,1]]]
Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPageEmpty5}
\begin{paste}{ugxProblemLinSysPageEmpty5}{ugxProblemLinSysPagePatch5}
\pastebutton{ugxProblemLinSysPageEmpty5}{\showpaste}
\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPagePatch6}
\begin{paste}{ugxProblemLinSysPageFull6}{ugxProblemLinSysPageEmpty6}
\pastebutton{ugxProblemLinSysPageFull6}{\hidepaste}
\tab{5}\spadcommand{nullSpace([[1,2,3],[2,3,4],[3,4,5]])}
\indentrel{3}\begin{verbatim}
   (6)  [[1,- 2,1]]
                              Type: List Vector Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLinSysPageEmpty6}
\begin{paste}{ugxProblemLinSysPageEmpty6}{ugxProblemLinSysPagePatch6}
\pastebutton{ugxProblemLinSysPageEmpty6}{\showpaste}
\tab{5}\spadcommand{nullSpace([[1,2,3],[2,3,4],[3,4,5]])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPagePatch1}
\begin{paste}{ugxProblemPolSysPageFull1}{ugxProblemPolSysPageEmpty1}
\pastebutton{ugxProblemPolSysPageFull1}{\hidepaste}
\tab{5}\spadcommand{solve([3*x**3 + y + 1,y**2 -4],[x,y])}
\indentrel{3}\begin{verbatim}
   (1)
                     2
   [[x= - 1,y= 2], [x  - x + 1= 0,y= 2],
       3
    [3x  - 1= 0,y= - 2]]
   Type: List List Equation Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPageEmpty1}
\begin{paste}{ugxProblemPolSysPageEmpty1}{ugxProblemPolSysPagePatch1}
\pastebutton{ugxProblemPolSysPageEmpty1}{\showpaste}
\tab{5}\spadcommand{solve([3*x**3 + y + 1,y**2 -4],[x,y])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPagePatch2}
\begin{paste}{ugxProblemPolSysPageFull2}{ugxProblemPolSysPageEmpty2}
\pastebutton{ugxProblemPolSysPageFull2}{\hidepaste}
\tab{5}\spadcommand{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])}
\indentrel{3}\begin{verbatim}
   (2)
               2
        z    3z  + z + 9   4     3      2
   [[x= Ä,y= ÄÄÄÄÄÄÄÄÄÄÄ,9z  + 6z  + 55z  + 15z - 90= 0]]
        3         3
   Type: List List Equation Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPageEmpty2}
\begin{paste}{ugxProblemPolSysPageEmpty2}{ugxProblemPolSysPagePatch2}
\pastebutton{ugxProblemPolSysPageEmpty2}{\showpaste}
\tab{5}\spadcommand{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPagePatch3}
\begin{paste}{ugxProblemPolSysPageFull3}{ugxProblemPolSysPageEmpty3}
\pastebutton{ugxProblemPolSysPageFull3}{\hidepaste}
\tab{5}\spadcommand{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])}
\indentrel{3}\begin{verbatim}
   (3)
         ÚÄÄÄ¿                   ÚÄÄÄ¿
        \³- 3  + 1            - \³- 3  + 1
   [[x= ÄÄÄÄÄÄÄÄÄÄ,y= 2], [x= ÄÄÄÄÄÄÄÄÄÄÄÄ,y= 2],
             2                      2
           ÚÄÄÄ¿ ÚÄ¿
        - \³- 1 \³3  - 1
    [x= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,y= - 2],
               3ÚÄ¿
              2\³3
         ÚÄÄÄ¿ ÚÄ¿
        \³- 1 \³3  - 1                1
    [x= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,y= - 2], [x= ÄÄÄÄ,y= - 2],
              3ÚÄ¿                  3ÚÄ¿
             2\³3                   \³3
    [x= - 1,y= 2]]
            Type: List List Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPageEmpty3}
\begin{paste}{ugxProblemPolSysPageEmpty3}{ugxProblemPolSysPagePatch3}
\pastebutton{ugxProblemPolSysPageEmpty3}{\showpaste}
\tab{5}\spadcommand{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])}
\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPagePatch4}
\begin{paste}{ugxProblemPolSysPageFull4}{ugxProblemPolSysPageEmpty4}
\pastebutton{ugxProblemPolSysPageFull4}{\hidepaste}
\tab{5}\spadcommand{solve([x**2*y - 1,x*y**2 - 2],.01)}
\indentrel{3}\begin{verbatim}
   (4)  [[y= 1.5859375,x= 0.79296875]]
              Type: List List Equation Polynomial Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPageEmpty4}
\begin{paste}{ugxProblemPolSysPageEmpty4}{ugxProblemPolSysPagePatch4}
\pastebutton{ugxProblemPolSysPageEmpty4}{\showpaste}
\tab{5}\spadcommand{solve([x**2*y - 1,x*y**2 - 2],.01)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPagePatch5}
\begin{paste}{ugxProblemPolSysPageFull5}{ugxProblemPolSysPageEmpty5}
\pastebutton{ugxProblemPolSysPageFull5}{\hidepaste}
\tab{5}\spadcommand{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)}
\indentrel{3}\begin{verbatim}
   (5)
        1625    1625
   [[y= ÄÄÄÄ,x= ÄÄÄÄ],
        1024    2048

           435445573689   1407
     [y= - ÄÄÄÄÄÄÄÄÄÄÄÄ - ÄÄÄÄ %i,
           549755813888   1024
            435445573689   1407
      x= - ÄÄÄÄÄÄÄÄÄÄÄÄÄ - ÄÄÄÄ %i]
           1099511627776   2048
     ,

           435445573689   1407
     [y= - ÄÄÄÄÄÄÄÄÄÄÄÄ + ÄÄÄÄ %i,
           549755813888   1024
            435445573689   1407
      x= - ÄÄÄÄÄÄÄÄÄÄÄÄÄ + ÄÄÄÄ %i]
           1099511627776   2048
     ]
Type: List List Equation Polynomial Complex Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPageEmpty5}
\begin{paste}{ugxProblemPolSysPageEmpty5}{ugxProblemPolSysPagePatch5}
\pastebutton{ugxProblemPolSysPageEmpty5}{\showpaste}
\tab{5}\spadcommand{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPagePatch6}
\begin{paste}{ugxProblemPolSysPageFull6}{ugxProblemPolSysPageEmpty6}
\pastebutton{ugxProblemPolSysPageFull6}{\hidepaste}
\tab{5}\spadcommand{solve([x**2/a = a,a = a*x],.001)}
\indentrel{3}\begin{verbatim}
   (6)  [[x= 1.0,a= - 1.0],[x= 1.0,a= 1.0]]
              Type: List List Equation Polynomial Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPageEmpty6}
\begin{paste}{ugxProblemPolSysPageEmpty6}{ugxProblemPolSysPagePatch6}
\pastebutton{ugxProblemPolSysPageEmpty6}{\showpaste}
\tab{5}\spadcommand{solve([x**2/a = a,a = a*x],.001)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPagePatch7}
\begin{paste}{ugxProblemPolSysPageFull7}{ugxProblemPolSysPageEmpty7}
\pastebutton{ugxProblemPolSysPageFull7}{\hidepaste}
\tab{5}\spadcommand{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])}
\indentrel{3}\begin{verbatim}
   (7)
           ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
           ³   4     3     2
           ³- a  + 2a  + 3a  + 3a + 1     - a - 1
   [[x= -  ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ,y= ÄÄÄÄÄÄÄ],
           ³             2                   a
          \³            a
         ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
         ³   4     3     2
         ³- a  + 2a  + 3a  + 3a + 1     - a - 1
    [x=  ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ,y= ÄÄÄÄÄÄÄ]]
         ³             2                   a
        \³            a
            Type: List List Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemPolSysPageEmpty7}
\begin{paste}{ugxProblemPolSysPageEmpty7}{ugxProblemPolSysPagePatch7}
\pastebutton{ugxProblemPolSysPageEmpty7}{\showpaste}
\tab{5}\spadcommand{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorFFPagePatch1}
\begin{paste}{ugProblemFactorFFPageFull1}{ugProblemFactorFFPageEmpty1}
\pastebutton{ugProblemFactorFFPageFull1}{\hidepaste}
\tab{5}\spadcommand{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18\bound{u }}
\indentrel{3}\begin{verbatim}
          4     2
   (1)  3x  + 2x  + 15x + 18
                         Type: Polynomial PrimeField 19
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorFFPageEmpty1}
\begin{paste}{ugProblemFactorFFPageEmpty1}{ugProblemFactorFFPagePatch1}
\pastebutton{ugProblemFactorFFPageEmpty1}{\showpaste}
\tab{5}\spadcommand{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18\bound{u }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorFFPagePatch2}
\begin{paste}{ugProblemFactorFFPageFull2}{ugProblemFactorFFPageEmpty2}
\pastebutton{ugProblemFactorFFPageFull2}{\hidepaste}
\tab{5}\spadcommand{factor u\free{u }}
\indentrel{3}\begin{verbatim}
                   3    2
   (2)  3(x + 18)(x  + x  + 8x + 13)
                Type: Factored Polynomial PrimeField 19
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorFFPageEmpty2}
\begin{paste}{ugProblemFactorFFPageEmpty2}{ugProblemFactorFFPagePatch2}
\pastebutton{ugProblemFactorFFPageEmpty2}{\showpaste}
\tab{5}\spadcommand{factor u\free{u }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorFFPagePatch3}
\begin{paste}{ugProblemFactorFFPageFull3}{ugProblemFactorFFPageEmpty3}
\pastebutton{ugProblemFactorFFPageFull3}{\hidepaste}
\tab{5}\spadcommand{factor(u :: POLY FFX(PF 19,3))\free{u }}
\indentrel{3}\begin{verbatim}
   (3)
                       2
     3(x + 18)(x + 5%BC  + 3%BC + 13)
  *
               2                        2
     (x + 16%BC  + 14%BC + 13)(x + 17%BC  + 2%BC + 13)
Type: Factored Polynomial FiniteFieldExtension(PrimeField 19,3)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorFFPageEmpty3}
\begin{paste}{ugProblemFactorFFPageEmpty3}{ugProblemFactorFFPagePatch3}
\pastebutton{ugProblemFactorFFPageEmpty3}{\showpaste}
\tab{5}\spadcommand{factor(u :: POLY FFX(PF 19,3))\free{u }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPagePatch1}
\begin{paste}{ugxProblemSeriesArithmeticPageFull1}{ugxProblemSeriesArithmeticPageEmpty1}
\pastebutton{ugxProblemSeriesArithmeticPageFull1}{\hidepaste}
\tab{5}\spadcommand{x := series x\bound{x }}
\indentrel{3}\begin{verbatim}
   (1)  x
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPageEmpty1}
\begin{paste}{ugxProblemSeriesArithmeticPageEmpty1}{ugxProblemSeriesArithmeticPagePatch1}
\pastebutton{ugxProblemSeriesArithmeticPageEmpty1}{\showpaste}
\tab{5}\spadcommand{x := series x\bound{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPagePatch2}
\begin{paste}{ugxProblemSeriesArithmeticPageFull2}{ugxProblemSeriesArithmeticPageEmpty2}
\pastebutton{ugxProblemSeriesArithmeticPageFull2}{\hidepaste}
\tab{5}\spadcommand{(3 + x) / (1 + 7*x)}
\indentrel{3}\begin{verbatim}
   (2)
                   2       3        4         5          6
     3 - 20x + 140x  - 980x  + 6860x  - 48020x  + 336140x
   + 
               7            8             9             10
     - 2352980x  + 16470860x  - 115296020x  + 807072140x
   + 
        11
     O(x  )
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPageEmpty2}
\begin{paste}{ugxProblemSeriesArithmeticPageEmpty2}{ugxProblemSeriesArithmeticPagePatch2}
\pastebutton{ugxProblemSeriesArithmeticPageEmpty2}{\showpaste}
\tab{5}\spadcommand{(3 + x) / (1 + 7*x)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPagePatch3}
\begin{paste}{ugxProblemSeriesArithmeticPageFull3}{ugxProblemSeriesArithmeticPageEmpty3}
\pastebutton{ugxProblemSeriesArithmeticPageFull3}{\hidepaste}
\tab{5}\spadcommand{base := 1 / (1 - x)\free{x }\bound{base }}
\indentrel{3}\begin{verbatim}
   (3)
              2    3    4    5    6    7    8    9    10
     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
   + 
        11
     O(x  )
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPageEmpty3}
\begin{paste}{ugxProblemSeriesArithmeticPageEmpty3}{ugxProblemSeriesArithmeticPagePatch3}
\pastebutton{ugxProblemSeriesArithmeticPageEmpty3}{\showpaste}
\tab{5}\spadcommand{base := 1 / (1 - x)\free{x }\bound{base }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPagePatch4}
\begin{paste}{ugxProblemSeriesArithmeticPageFull4}{ugxProblemSeriesArithmeticPageEmpty4}
\pastebutton{ugxProblemSeriesArithmeticPageFull4}{\hidepaste}
\tab{5}\spadcommand{expon := x * base\free{x base }\bound{expon }}
\indentrel{3}\begin{verbatim}
   (4)
          2    3    4    5    6    7    8    9    10    11
     x + x  + x  + x  + x  + x  + x  + x  + x  + x   + x
   + 
        12
     O(x  )
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPageEmpty4}
\begin{paste}{ugxProblemSeriesArithmeticPageEmpty4}{ugxProblemSeriesArithmeticPagePatch4}
\pastebutton{ugxProblemSeriesArithmeticPageEmpty4}{\showpaste}
\tab{5}\spadcommand{expon := x * base\free{x base }\bound{expon }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPagePatch5}
\begin{paste}{ugxProblemSeriesArithmeticPageFull5}{ugxProblemSeriesArithmeticPageEmpty5}
\pastebutton{ugxProblemSeriesArithmeticPageFull5}{\hidepaste}
\tab{5}\spadcommand{base ** expon\free{base expon }}
\indentrel{3}\begin{verbatim}
   (5)
          2   3  3   7  4   43  5   649  6   241  7
     1 + x  + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
              2      3      12      120       30
   + 
     3706  8   85763  9   245339  10      11
     ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x   + O(x  )
      315       5040       10080
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesArithmeticPageEmpty5}
\begin{paste}{ugxProblemSeriesArithmeticPageEmpty5}{ugxProblemSeriesArithmeticPagePatch5}
\pastebutton{ugxProblemSeriesArithmeticPageEmpty5}{\showpaste}
\tab{5}\spadcommand{base ** expon\free{base expon }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch1}
\begin{paste}{ugxProblemFiniteConversionPageFull1}{ugxProblemFiniteConversionPageEmpty1}
\pastebutton{ugxProblemFiniteConversionPageFull1}{\hidepaste}
\tab{5}\spadcommand{K := PrimeField 3\bound{K }}
\indentrel{3}\begin{verbatim}
   (1)  PrimeField 3
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty1}
\begin{paste}{ugxProblemFiniteConversionPageEmpty1}{ugxProblemFiniteConversionPagePatch1}
\pastebutton{ugxProblemFiniteConversionPageEmpty1}{\showpaste}
\tab{5}\spadcommand{K := PrimeField 3\bound{K }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch2}
\begin{paste}{ugxProblemFiniteConversionPageFull2}{ugxProblemFiniteConversionPageEmpty2}
\pastebutton{ugxProblemFiniteConversionPageFull2}{\hidepaste}
\tab{5}\spadcommand{(m,n) := (4,8)\bound{m n }}
\indentrel{3}\begin{verbatim}
   (2)  8
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty2}
\begin{paste}{ugxProblemFiniteConversionPageEmpty2}{ugxProblemFiniteConversionPagePatch2}
\pastebutton{ugxProblemFiniteConversionPageEmpty2}{\showpaste}
\tab{5}\spadcommand{(m,n) := (4,8)\bound{m n }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch3}
\begin{paste}{ugxProblemFiniteConversionPageFull3}{ugxProblemFiniteConversionPageEmpty3}
\pastebutton{ugxProblemFiniteConversionPageFull3}{\hidepaste}
\tab{5}\spadcommand{Km := FiniteFieldExtension(K,m)\bound{Km }\free{K m }}
\indentrel{3}\begin{verbatim}
   (3)  FiniteFieldExtension(PrimeField 3,4)
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty3}
\begin{paste}{ugxProblemFiniteConversionPageEmpty3}{ugxProblemFiniteConversionPagePatch3}
\pastebutton{ugxProblemFiniteConversionPageEmpty3}{\showpaste}
\tab{5}\spadcommand{Km := FiniteFieldExtension(K,m)\bound{Km }\free{K m }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch4}
\begin{paste}{ugxProblemFiniteConversionPageFull4}{ugxProblemFiniteConversionPageEmpty4}
\pastebutton{ugxProblemFiniteConversionPageFull4}{\hidepaste}
\tab{5}\spadcommand{Kn := FiniteFieldExtension(K,n)\bound{Kn }\free{K n }}
\indentrel{3}\begin{verbatim}
   (4)  FiniteFieldExtension(PrimeField 3,8)
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty4}
\begin{paste}{ugxProblemFiniteConversionPageEmpty4}{ugxProblemFiniteConversionPagePatch4}
\pastebutton{ugxProblemFiniteConversionPageEmpty4}{\showpaste}
\tab{5}\spadcommand{Kn := FiniteFieldExtension(K,n)\bound{Kn }\free{K n }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch5}
\begin{paste}{ugxProblemFiniteConversionPageFull5}{ugxProblemFiniteConversionPageEmpty5}
\pastebutton{ugxProblemFiniteConversionPageFull5}{\hidepaste}
\tab{5}\spadcommand{a1 := random()$Km\bound{a1 }\free{Km }}
\indentrel{3}\begin{verbatim}
            3       2
   (5)  2%BD  + 2%BD  + 2%BD + 1
             Type: FiniteFieldExtension(PrimeField 3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty5}
\begin{paste}{ugxProblemFiniteConversionPageEmpty5}{ugxProblemFiniteConversionPagePatch5}
\pastebutton{ugxProblemFiniteConversionPageEmpty5}{\showpaste}
\tab{5}\spadcommand{a1 := random()$Km\bound{a1 }\free{Km }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch6}
\begin{paste}{ugxProblemFiniteConversionPageFull6}{ugxProblemFiniteConversionPageEmpty6}
\pastebutton{ugxProblemFiniteConversionPageFull6}{\hidepaste}
\tab{5}\spadcommand{b1 := random()$Km\bound{b1 }\free{Km }}
\indentrel{3}\begin{verbatim}
           3      2
   (6)  %BD  + %BD  + 2%BD + 2
             Type: FiniteFieldExtension(PrimeField 3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty6}
\begin{paste}{ugxProblemFiniteConversionPageEmpty6}{ugxProblemFiniteConversionPagePatch6}
\pastebutton{ugxProblemFiniteConversionPageEmpty6}{\showpaste}
\tab{5}\spadcommand{b1 := random()$Km\bound{b1 }\free{Km }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch7}
\begin{paste}{ugxProblemFiniteConversionPageFull7}{ugxProblemFiniteConversionPageEmpty7}
\pastebutton{ugxProblemFiniteConversionPageFull7}{\hidepaste}
\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a2 }\free{a1 Kn }}
\indentrel{3}\begin{verbatim}
            6
   (7)  2%BE  + 1
             Type: FiniteFieldExtension(PrimeField 3,8)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty7}
\begin{paste}{ugxProblemFiniteConversionPageEmpty7}{ugxProblemFiniteConversionPagePatch7}
\pastebutton{ugxProblemFiniteConversionPageEmpty7}{\showpaste}
\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a2 }\free{a1 Kn }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch8}
\begin{paste}{ugxProblemFiniteConversionPageFull8}{ugxProblemFiniteConversionPageEmpty8}
\pastebutton{ugxProblemFiniteConversionPageFull8}{\hidepaste}
\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b2 }\free{b1 Kn }}
\indentrel{3}\begin{verbatim}
            6       4      2
   (8)  2%BE  + 2%BE  + %BE  + 2
             Type: FiniteFieldExtension(PrimeField 3,8)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty8}
\begin{paste}{ugxProblemFiniteConversionPageEmpty8}{ugxProblemFiniteConversionPagePatch8}
\pastebutton{ugxProblemFiniteConversionPageEmpty8}{\showpaste}
\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b2 }\free{b1 Kn }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch9}
\begin{paste}{ugxProblemFiniteConversionPageFull9}{ugxProblemFiniteConversionPageEmpty9}
\pastebutton{ugxProblemFiniteConversionPageFull9}{\hidepaste}
\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a1 a2 b1 b2 Km Kn }}
\indentrel{3}\begin{verbatim}
   (9)  0
             Type: FiniteFieldExtension(PrimeField 3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty9}
\begin{paste}{ugxProblemFiniteConversionPageEmpty9}{ugxProblemFiniteConversionPagePatch9}
\pastebutton{ugxProblemFiniteConversionPageEmpty9}{\showpaste}
\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a1 a2 b1 b2 Km Kn }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch10}
\begin{paste}{ugxProblemFiniteConversionPageFull10}{ugxProblemFiniteConversionPageEmpty10}
\pastebutton{ugxProblemFiniteConversionPageFull10}{\hidepaste}
\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a1 a2 b1 b2 Km Kn }}
\indentrel{3}\begin{verbatim}
   (10)  0
             Type: FiniteFieldExtension(PrimeField 3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty10}
\begin{paste}{ugxProblemFiniteConversionPageEmpty10}{ugxProblemFiniteConversionPagePatch10}
\pastebutton{ugxProblemFiniteConversionPageEmpty10}{\showpaste}
\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a1 a2 b1 b2 Km Kn }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch11}
\begin{paste}{ugxProblemFiniteConversionPageFull11}{ugxProblemFiniteConversionPageEmpty11}
\pastebutton{ugxProblemFiniteConversionPageFull11}{\hidepaste}
\tab{5}\spadcommand{Km := FFCGX(K,m)\bound{Km2 }\free{K m }}
\indentrel{3}\begin{verbatim}
   (11)  FiniteFieldCyclicGroupExtension(PrimeField 3,4)
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty11}
\begin{paste}{ugxProblemFiniteConversionPageEmpty11}{ugxProblemFiniteConversionPagePatch11}
\pastebutton{ugxProblemFiniteConversionPageEmpty11}{\showpaste}
\tab{5}\spadcommand{Km := FFCGX(K,m)\bound{Km2 }\free{K m }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch12}
\begin{paste}{ugxProblemFiniteConversionPageFull12}{ugxProblemFiniteConversionPageEmpty12}
\pastebutton{ugxProblemFiniteConversionPageFull12}{\hidepaste}
\tab{5}\spadcommand{Kn := FFNBX(K,n)\bound{Kn2 }\free{K n }}
\indentrel{3}\begin{verbatim}
   (12)  FiniteFieldNormalBasisExtension(PrimeField 3,8)
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty12}
\begin{paste}{ugxProblemFiniteConversionPageEmpty12}{ugxProblemFiniteConversionPagePatch12}
\pastebutton{ugxProblemFiniteConversionPageEmpty12}{\showpaste}
\tab{5}\spadcommand{Kn := FFNBX(K,n)\bound{Kn2 }\free{K n }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch13}
\begin{paste}{ugxProblemFiniteConversionPageFull13}{ugxProblemFiniteConversionPageEmpty13}
\pastebutton{ugxProblemFiniteConversionPageFull13}{\hidepaste}
\tab{5}\spadcommand{(a1,b1) := (random()$Km,random()$Km)\bound{a12 b12 }\free{Km2 }}
\indentrel{3}\begin{verbatim}
            13
   (13)  %BF
  Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty13}
\begin{paste}{ugxProblemFiniteConversionPageEmpty13}{ugxProblemFiniteConversionPagePatch13}
\pastebutton{ugxProblemFiniteConversionPageEmpty13}{\showpaste}
\tab{5}\spadcommand{(a1,b1) := (random()$Km,random()$Km)\bound{a12 b12 }\free{Km2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch14}
\begin{paste}{ugxProblemFiniteConversionPageFull14}{ugxProblemFiniteConversionPageEmpty14}
\pastebutton{ugxProblemFiniteConversionPageFull14}{\hidepaste}
\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a22 }\free{a12 Kn2 }}
\indentrel{3}\begin{verbatim}
   (14)
        6        5        4        2
       q        q        q        q        q
   2%BG   + 2%BG   + 2%BG   + 2%BG   + 2%BG  + 2%BG
  Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty14}
\begin{paste}{ugxProblemFiniteConversionPageEmpty14}{ugxProblemFiniteConversionPagePatch14}
\pastebutton{ugxProblemFiniteConversionPageEmpty14}{\showpaste}
\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a22 }\free{a12 Kn2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch15}
\begin{paste}{ugxProblemFiniteConversionPageFull15}{ugxProblemFiniteConversionPageEmpty15}
\pastebutton{ugxProblemFiniteConversionPageFull15}{\hidepaste}
\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b22 }\free{b12 Kn2 }}
\indentrel{3}\begin{verbatim}
   (15)
          7       6       5       4        3       2
         q       q       q       q        q       q
     2%BG   + %BG   + %BG   + %BG   + 2%BG   + %BG
   + 
        q
     %BG  + %BG
  Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty15}
\begin{paste}{ugxProblemFiniteConversionPageEmpty15}{ugxProblemFiniteConversionPagePatch15}
\pastebutton{ugxProblemFiniteConversionPageEmpty15}{\showpaste}
\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b22 }\free{b12 Kn2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch16}
\begin{paste}{ugxProblemFiniteConversionPageFull16}{ugxProblemFiniteConversionPageEmpty16}
\pastebutton{ugxProblemFiniteConversionPageFull16}{\hidepaste}
\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a12 a22 b12 b22 Km2 }}
\indentrel{3}\begin{verbatim}
   (16)  0
  Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty16}
\begin{paste}{ugxProblemFiniteConversionPageEmpty16}{ugxProblemFiniteConversionPagePatch16}
\pastebutton{ugxProblemFiniteConversionPageEmpty16}{\showpaste}
\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a12 a22 b12 b22 Km2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPagePatch17}
\begin{paste}{ugxProblemFiniteConversionPageFull17}{ugxProblemFiniteConversionPageEmpty17}
\pastebutton{ugxProblemFiniteConversionPageFull17}{\hidepaste}
\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a12 a22 b12 b22 Km2 }}
\indentrel{3}\begin{verbatim}
   (17)  0
  Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteConversionPageEmpty17}
\begin{paste}{ugxProblemFiniteConversionPageEmpty17}{ugxProblemFiniteConversionPagePatch17}
\pastebutton{ugxProblemFiniteConversionPageEmpty17}{\showpaste}
\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a12 a22 b12 b22 Km2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesSubstitutePagePatch1}
\begin{paste}{ugxProblemSeriesSubstitutePageFull1}{ugxProblemSeriesSubstitutePageEmpty1}
\pastebutton{ugxProblemSeriesSubstitutePageFull1}{\hidepaste}
\tab{5}\spadcommand{f := taylor(exp(x))\bound{f }}
\indentrel{3}\begin{verbatim}
   (1)
             1  2   1  3    1  4    1   5    1   6
     1 + x + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
             2      6      24      120      720
   + 
     1   7     1    8      1    9      1     10      11
   ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x   + O(x  )
   5040      40320      362880      3628800
   Type: UnivariateTaylorSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesSubstitutePageEmpty1}
\begin{paste}{ugxProblemSeriesSubstitutePageEmpty1}{ugxProblemSeriesSubstitutePagePatch1}
\pastebutton{ugxProblemSeriesSubstitutePageEmpty1}{\showpaste}
\tab{5}\spadcommand{f := taylor(exp(x))\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesSubstitutePagePatch2}
\begin{paste}{ugxProblemSeriesSubstitutePageFull2}{ugxProblemSeriesSubstitutePageEmpty2}
\pastebutton{ugxProblemSeriesSubstitutePageFull2}{\hidepaste}
\tab{5}\spadcommand{eval(f,1.0)}
\indentrel{3}\begin{verbatim}
   (2)
   [1.0, 2.0, 2.5, 2.6666666666 666666667,
    2.7083333333 333333333, 2.7166666666 666666667,
    2.7180555555 555555556, 2.7182539682 53968254,
    2.7182787698 412698413, 2.7182815255 731922399, ...]
                          Type: Stream Expression Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesSubstitutePageEmpty2}
\begin{paste}{ugxProblemSeriesSubstitutePageEmpty2}{ugxProblemSeriesSubstitutePagePatch2}
\pastebutton{ugxProblemSeriesSubstitutePageEmpty2}{\showpaste}
\tab{5}\spadcommand{eval(f,1.0)}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch1}
\begin{paste}{ugProblemFactorAlgPageFull1}{ugProblemFactorAlgPageEmpty1}
\pastebutton{ugProblemFactorAlgPageFull1}{\hidepaste}
\tab{5}\spadcommand{aa := rootOf(aa**2+aa+1)\bound{aa }}
\indentrel{3}\begin{verbatim}
   (1)  aa
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty1}
\begin{paste}{ugProblemFactorAlgPageEmpty1}{ugProblemFactorAlgPagePatch1}
\pastebutton{ugProblemFactorAlgPageEmpty1}{\showpaste}
\tab{5}\spadcommand{aa := rootOf(aa**2+aa+1)\bound{aa }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch2}
\begin{paste}{ugProblemFactorAlgPageFull2}{ugProblemFactorAlgPageEmpty2}
\pastebutton{ugProblemFactorAlgPageFull2}{\hidepaste}
\tab{5}\spadcommand{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2\free{aa }\bound{p }}
\indentrel{3}\begin{verbatim}
   (2)
                5               3         4
     (- aa - 1)y  + ((- aa - 1)x  + aa x)y
   + 
                  2                 3
     ((- 2aa - 2)x  + (- 2aa - 2)x)y
   + 
                  5               4        3        2  2
     ((- 2aa - 2)x  + (- 2aa - 2)x  + 2aa x  + 2aa x )y
   + 
                 4               3              2
     ((- aa - 1)x  + (- 2aa - 2)x  + (- aa - 1)x )y
   + 
                7               6    5        4       3
     (- aa - 1)x  + (- 2aa - 2)x  - x  + 2aa x  + aa x
                       Type: Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty2}
\begin{paste}{ugProblemFactorAlgPageEmpty2}{ugProblemFactorAlgPagePatch2}
\pastebutton{ugProblemFactorAlgPageEmpty2}{\showpaste}
\tab{5}\spadcommand{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2\free{aa }\bound{p }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch3}
\begin{paste}{ugProblemFactorAlgPageFull3}{ugProblemFactorAlgPageEmpty3}
\pastebutton{ugProblemFactorAlgPageFull3}{\hidepaste}
\tab{5}\spadcommand{factor(p,[aa])\free{p aa }}
\indentrel{3}\begin{verbatim}
                        3                 2    2     2
   (3)  (- aa - 1)(y + x  + (- aa - 1)x)(y  + x  + x)
              Type: Factored Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty3}
\begin{paste}{ugProblemFactorAlgPageEmpty3}{ugProblemFactorAlgPagePatch3}
\pastebutton{ugProblemFactorAlgPageEmpty3}{\showpaste}
\tab{5}\spadcommand{factor(p,[aa])\free{p aa }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch4}
\begin{paste}{ugProblemFactorAlgPageFull4}{ugProblemFactorAlgPageEmpty4}
\pastebutton{ugProblemFactorAlgPageFull4}{\hidepaste}
\tab{5}\spadcommand{factor(x**2+3)}
\indentrel{3}\begin{verbatim}
         2
   (4)  x  + 3
                      Type: Factored Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty4}
\begin{paste}{ugProblemFactorAlgPageEmpty4}{ugProblemFactorAlgPagePatch4}
\pastebutton{ugProblemFactorAlgPageEmpty4}{\showpaste}
\tab{5}\spadcommand{factor(x**2+3)}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch5}
\begin{paste}{ugProblemFactorAlgPageFull5}{ugProblemFactorAlgPageEmpty5}
\pastebutton{ugProblemFactorAlgPageFull5}{\hidepaste}
\tab{5}\spadcommand{factor(x**2+3,[aa])\free{aa }}
\indentrel{3}\begin{verbatim}
   (5)  (x - 2aa - 1)(x + 2aa + 1)
              Type: Factored Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty5}
\begin{paste}{ugProblemFactorAlgPageEmpty5}{ugProblemFactorAlgPagePatch5}
\pastebutton{ugProblemFactorAlgPageEmpty5}{\showpaste}
\tab{5}\spadcommand{factor(x**2+3,[aa])\free{aa }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch6}
\begin{paste}{ugProblemFactorAlgPageFull6}{ugProblemFactorAlgPageEmpty6}
\pastebutton{ugProblemFactorAlgPageFull6}{\hidepaste}
\tab{5}\spadcommand{factor(x**6+108,[aa])\free{aa }}
\indentrel{3}\begin{verbatim}
          3              3
   (6)  (x  - 12aa - 6)(x  + 12aa + 6)
              Type: Factored Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty6}
\begin{paste}{ugProblemFactorAlgPageEmpty6}{ugProblemFactorAlgPagePatch6}
\pastebutton{ugProblemFactorAlgPageEmpty6}{\showpaste}
\tab{5}\spadcommand{factor(x**6+108,[aa])\free{aa }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch7}
\begin{paste}{ugProblemFactorAlgPageFull7}{ugProblemFactorAlgPageEmpty7}
\pastebutton{ugProblemFactorAlgPageFull7}{\hidepaste}
\tab{5}\spadcommand{bb:=rootOf(bb**3-2)\bound{bb }}
\indentrel{3}\begin{verbatim}
   (7)  bb
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty7}
\begin{paste}{ugProblemFactorAlgPageEmpty7}{ugProblemFactorAlgPagePatch7}
\pastebutton{ugProblemFactorAlgPageEmpty7}{\showpaste}
\tab{5}\spadcommand{bb:=rootOf(bb**3-2)\bound{bb }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch8}
\begin{paste}{ugProblemFactorAlgPageFull8}{ugProblemFactorAlgPageEmpty8}
\pastebutton{ugProblemFactorAlgPageFull8}{\hidepaste}
\tab{5}\spadcommand{factor(x**6+108,[bb])\free{bb }}
\indentrel{3}\begin{verbatim}
   (8)
     2              2   2      2   2              2
   (x  - 3bb x + 3bb )(x  + 3bb )(x  + 3bb x + 3bb )
              Type: Factored Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty8}
\begin{paste}{ugProblemFactorAlgPageEmpty8}{ugProblemFactorAlgPagePatch8}
\pastebutton{ugProblemFactorAlgPageEmpty8}{\showpaste}
\tab{5}\spadcommand{factor(x**6+108,[bb])\free{bb }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPagePatch9}
\begin{paste}{ugProblemFactorAlgPageFull9}{ugProblemFactorAlgPageEmpty9}
\pastebutton{ugProblemFactorAlgPageFull9}{\hidepaste}
\tab{5}\spadcommand{factor(x**6+108,[aa,bb])\free{aa bb }}
\indentrel{3}\begin{verbatim}
   (9)
     (x + (- 2aa - 1)bb)(x + (- aa - 2)bb)
  *
     (x + (- aa + 1)bb)(x + (aa - 1)bb)(x + (aa + 2)bb)
  *
     (x + (2aa + 1)bb)
              Type: Factored Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorAlgPageEmpty9}
\begin{paste}{ugProblemFactorAlgPageEmpty9}{ugProblemFactorAlgPagePatch9}
\pastebutton{ugProblemFactorAlgPageEmpty9}{\showpaste}
\tab{5}\spadcommand{factor(x**6+108,[aa,bb])\free{aa bb }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch1}
\begin{paste}{ugxProblemSeriesConversionsPageFull1}{ugxProblemSeriesConversionsPageEmpty1}
\pastebutton{ugxProblemSeriesConversionsPageFull1}{\hidepaste}
\tab{5}\spadcommand{taylor(sin(x),x = 0)}
\indentrel{3}\begin{verbatim}
            1  3    1   5     1   7      1    9      11
   (1)  x - Ä x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x  + O(x  )
            6      120      5040      362880
   Type: UnivariateTaylorSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty1}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty1}{ugxProblemSeriesConversionsPagePatch1}
\pastebutton{ugxProblemSeriesConversionsPageEmpty1}{\showpaste}
\tab{5}\spadcommand{taylor(sin(x),x = 0)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch2}
\begin{paste}{ugxProblemSeriesConversionsPageFull2}{ugxProblemSeriesConversionsPageEmpty2}
\pastebutton{ugxProblemSeriesConversionsPageFull2}{\hidepaste}
\tab{5}\spadcommand{taylor(sin(x),x = \%pi/6)}
\indentrel{3}\begin{verbatim}
   (2)
          ÚÄ¿                             ÚÄ¿
     1   \³3       %pi    1      %pi 2   \³3       %pi 3
     Ä + ÄÄÄÄ (x - ÄÄÄ) - Ä (x - ÄÄÄ)  - ÄÄÄÄ (x - ÄÄÄ)
     2     2        6     4       6       12        6
   + 
                      ÚÄ¿
      1      %pi 4   \³3       %pi 5     1       %pi 6
     ÄÄ (x - ÄÄÄ)  + ÄÄÄÄ (x - ÄÄÄ)  - ÄÄÄÄ (x - ÄÄÄ)
     48       6       240       6      1440       6
   + 
         ÚÄ¿
        \³3       %pi 7     1        %pi 8
     - ÄÄÄÄÄ (x - ÄÄÄ)  + ÄÄÄÄÄ (x - ÄÄÄ)
       10080       6      80640       6
   + 
       ÚÄ¿
      \³3        %pi 9      1         %pi 10
     ÄÄÄÄÄÄ (x - ÄÄÄ)  - ÄÄÄÄÄÄÄ (x - ÄÄÄ)
     725760       6      7257600       6
   + 
            %pi 11
     O((x - ÄÄÄ)  )
             6
Type: UnivariateTaylorSeries(Expression Integer,x,pi/6)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty2}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty2}{ugxProblemSeriesConversionsPagePatch2}
\pastebutton{ugxProblemSeriesConversionsPageEmpty2}{\showpaste}
\tab{5}\spadcommand{taylor(sin(x),x = \%pi/6)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch3}
\begin{paste}{ugxProblemSeriesConversionsPageFull3}{ugxProblemSeriesConversionsPageEmpty3}
\pastebutton{ugxProblemSeriesConversionsPageFull3}{\hidepaste}
\tab{5}\spadcommand{taylor(tan(x*y),x = 0)}
\indentrel{3}\begin{verbatim}
   (3)
          3        5         7         9
         y   3   2y   5   17y   7   62y   9      11
   y x + ÄÄ x  + ÄÄÄ x  + ÄÄÄÄ x  + ÄÄÄÄ x  + O(x  )
          3       15       315      2835
   Type: UnivariateTaylorSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty3}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty3}{ugxProblemSeriesConversionsPagePatch3}
\pastebutton{ugxProblemSeriesConversionsPageEmpty3}{\showpaste}
\tab{5}\spadcommand{taylor(tan(x*y),x = 0)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch4}
\begin{paste}{ugxProblemSeriesConversionsPageFull4}{ugxProblemSeriesConversionsPageEmpty4}
\pastebutton{ugxProblemSeriesConversionsPageFull4}{\hidepaste}
\tab{5}\spadcommand{taylor(tan(x*y),y = 0)}
\indentrel{3}\begin{verbatim}
   (4)
          3        5         7         9
         x   3   2x   5   17x   7   62x   9      11
   x y + ÄÄ y  + ÄÄÄ y  + ÄÄÄÄ y  + ÄÄÄÄ y  + O(y  )
          3       15       315      2835
   Type: UnivariateTaylorSeries(Expression Integer,y,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty4}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty4}{ugxProblemSeriesConversionsPagePatch4}
\pastebutton{ugxProblemSeriesConversionsPageEmpty4}{\showpaste}
\tab{5}\spadcommand{taylor(tan(x*y),y = 0)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch5}
\begin{paste}{ugxProblemSeriesConversionsPageFull5}{ugxProblemSeriesConversionsPageEmpty5}
\pastebutton{ugxProblemSeriesConversionsPageFull5}{\hidepaste}
\tab{5}\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0)\bound{bern }}
\indentrel{3}\begin{verbatim}
   (5)
                      2                 3     2
         2x - 1     6x  - 6x + 1  2   2x  - 3x  + x  3
     1 + ÄÄÄÄÄÄ t + ÄÄÄÄÄÄÄÄÄÄÄÄ t  + ÄÄÄÄÄÄÄÄÄÄÄÄÄ t
            2            12                 12
   + 
        4      3      2            5      4      3
     30x  - 60x  + 30x  - 1  4   6x  - 15x  + 10x  - x  5
     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t  + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
               720                        720
   + 
        6       5       4      2
     42x  - 126x  + 105x  - 21x  + 1  6
     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
                  30240
   + 
       7      6      5     3
     6x  - 21x  + 21x  - 7x  + x  7
     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
                30240
   + 
        8       7       6      4      2
     30x  - 120x  + 140x  - 70x  + 20x  - 1  8
     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
                     1209600
   + 
        9      8      7      5      3
     10x  - 45x  + 60x  - 42x  + 20x  - 3x  9
     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
                    3628800
   + 
        10       9       8       6       4      2
     66x   - 330x  + 495x  - 462x  + 330x  - 99x  + 5  10
     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
                         239500800
   + 
        11
     O(t  )
   Type: UnivariateTaylorSeries(Expression Integer,t,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty5}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty5}{ugxProblemSeriesConversionsPagePatch5}
\pastebutton{ugxProblemSeriesConversionsPageEmpty5}{\showpaste}
\tab{5}\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0)\bound{bern }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch6}
\begin{paste}{ugxProblemSeriesConversionsPageFull6}{ugxProblemSeriesConversionsPageEmpty6}
\pastebutton{ugxProblemSeriesConversionsPageFull6}{\hidepaste}
\tab{5}\spadcommand{factorial(6) * coefficient(bern,6)\free{bern }}
\indentrel{3}\begin{verbatim}
           6       5       4      2
        42x  - 126x  + 105x  - 21x  + 1
   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                       42
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty6}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty6}{ugxProblemSeriesConversionsPagePatch6}
\pastebutton{ugxProblemSeriesConversionsPageEmpty6}{\showpaste}
\tab{5}\spadcommand{factorial(6) * coefficient(bern,6)\free{bern }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch7}
\begin{paste}{ugxProblemSeriesConversionsPageFull7}{ugxProblemSeriesConversionsPageEmpty7}
\pastebutton{ugxProblemSeriesConversionsPageFull7}{\hidepaste}
\tab{5}\spadcommand{bernoulliB(6,x)}
\indentrel{3}\begin{verbatim}
         6     5   5  4   1  2    1
   (7)  x  - 3x  + Ä x  - Ä x  + ÄÄ
                   2      2      42
                      Type: Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty7}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty7}{ugxProblemSeriesConversionsPagePatch7}
\pastebutton{ugxProblemSeriesConversionsPageEmpty7}{\showpaste}
\tab{5}\spadcommand{bernoulliB(6,x)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch8}
\begin{paste}{ugxProblemSeriesConversionsPageFull8}{ugxProblemSeriesConversionsPageEmpty8}
\pastebutton{ugxProblemSeriesConversionsPageFull8}{\hidepaste}
\tab{5}\spadcommand{laurent(x/log(x),x = 1)}
\indentrel{3}\begin{verbatim}
   (8)
            - 1   3    5            1        2
     (x - 1)    + Ä + ÄÄ (x - 1) - ÄÄ (x - 1)
                  2   12           24
   + 
      11        3    11         4    271         5
     ÄÄÄ (x - 1)  - ÄÄÄÄ (x - 1)  + ÄÄÄÄÄ (x - 1)
     720            1440            60480
   + 
        13         6     7297         7     425         8
     - ÄÄÄÄ (x - 1)  + ÄÄÄÄÄÄÄ (x - 1)  - ÄÄÄÄÄÄ (x - 1)
       4480            3628800            290304
   + 
       530113         9            10
     ÄÄÄÄÄÄÄÄÄ (x - 1)  + O((x - 1)  )
     479001600
  Type: UnivariateLaurentSeries(Expression Integer,x,1)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty8}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty8}{ugxProblemSeriesConversionsPagePatch8}
\pastebutton{ugxProblemSeriesConversionsPageEmpty8}{\showpaste}
\tab{5}\spadcommand{laurent(x/log(x),x = 1)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch9}
\begin{paste}{ugxProblemSeriesConversionsPageFull9}{ugxProblemSeriesConversionsPageEmpty9}
\pastebutton{ugxProblemSeriesConversionsPageFull9}{\hidepaste}
\tab{5}\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)}
\indentrel{3}\begin{verbatim}
   (9)
                 1                3                 7
               - Ä                Ä                 Ä
          3%pi   2    1      3%pi 2    1       3%pi 2
     (x - ÄÄÄÄ)    + ÄÄ (x - ÄÄÄÄ)  + ÄÄÄ (x - ÄÄÄÄ)
            2        12        2      160        2
   + 
            3%pi 5
     O((x - ÄÄÄÄ) )
              2
Type: UnivariatePuiseuxSeries(Expression Integer,x,(3*pi)/2)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty9}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty9}{ugxProblemSeriesConversionsPagePatch9}
\pastebutton{ugxProblemSeriesConversionsPageEmpty9}{\showpaste}
\tab{5}\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPagePatch10}
\begin{paste}{ugxProblemSeriesConversionsPageFull10}{ugxProblemSeriesConversionsPageEmpty10}
\pastebutton{ugxProblemSeriesConversionsPageFull10}{\hidepaste}
\tab{5}\spadcommand{series(x**x,x=0)}
\indentrel{3}\begin{verbatim}
   (10)
                         2            3            4
                   log(x)   2   log(x)   3   log(x)   4
     1 + log(x)x + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x
                      2            6            24
   + 
           5            6            7            8
     log(x)   5   log(x)   6   log(x)   7   log(x)   8
     ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x
       120          720          5040        40320
   + 
           9            10
     log(x)   9   log(x)    10      11
     ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄÄ x   + O(x  )
      362880       3628800
Type: GeneralUnivariatePowerSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesConversionsPageEmpty10}
\begin{paste}{ugxProblemSeriesConversionsPageEmpty10}{ugxProblemSeriesConversionsPagePatch10}
\pastebutton{ugxProblemSeriesConversionsPageEmpty10}{\showpaste}
\tab{5}\spadcommand{series(x**x,x=0)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPagePatch1}
\begin{paste}{ugxProblemDEQSeriesPageFull1}{ugxProblemDEQSeriesPageEmpty1}
\pastebutton{ugxProblemDEQSeriesPageFull1}{\hidepaste}
\tab{5}\spadcommand{)set streams calculate 7\bound{c7 }}
\indentrel{3}\begin{verbatim}
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPageEmpty1}
\begin{paste}{ugxProblemDEQSeriesPageEmpty1}{ugxProblemDEQSeriesPagePatch1}
\pastebutton{ugxProblemDEQSeriesPageEmpty1}{\showpaste}
\tab{5}\spadcommand{)set streams calculate 7\bound{c7 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPagePatch2}
\begin{paste}{ugxProblemDEQSeriesPageFull2}{ugxProblemDEQSeriesPageEmpty2}
\pastebutton{ugxProblemDEQSeriesPageFull2}{\hidepaste}
\tab{5}\spadcommand{y := operator 'y\bound{y }}
\indentrel{3}\begin{verbatim}
   (1)  y
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPageEmpty2}
\begin{paste}{ugxProblemDEQSeriesPageEmpty2}{ugxProblemDEQSeriesPagePatch2}
\pastebutton{ugxProblemDEQSeriesPageEmpty2}{\showpaste}
\tab{5}\spadcommand{y := operator 'y\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPagePatch3}
\begin{paste}{ugxProblemDEQSeriesPageFull3}{ugxProblemDEQSeriesPageEmpty3}
\pastebutton{ugxProblemDEQSeriesPageFull3}{\hidepaste}
\tab{5}\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq }\free{y }}
\indentrel{3}\begin{verbatim}
         ,,,        y(x)     ,,
   (2)  y   (x) - %e    sin(y  (x))= cos(x)

                      Type: Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPageEmpty3}
\begin{paste}{ugxProblemDEQSeriesPageEmpty3}{ugxProblemDEQSeriesPagePatch3}
\pastebutton{ugxProblemDEQSeriesPageEmpty3}{\showpaste}
\tab{5}\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPagePatch4}
\begin{paste}{ugxProblemDEQSeriesPageFull4}{ugxProblemDEQSeriesPageEmpty4}
\pastebutton{ugxProblemDEQSeriesPageFull4}{\hidepaste}
\tab{5}\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y }\free{eq }\free{c7 }}
\indentrel{3}\begin{verbatim}
   (3)
                          2            3
         1  3   %e  4   %e  - 1  5   %e  - 2%e  6
     1 + Ä x  + ÄÄ x  + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄÄÄ x
         6      24        120           720
   + 
       4      2
     %e  - 8%e  + 4%e + 1  7      8
     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ x  + O(x )
             5040
   Type: UnivariateTaylorSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPageEmpty4}
\begin{paste}{ugxProblemDEQSeriesPageEmpty4}{ugxProblemDEQSeriesPagePatch4}
\pastebutton{ugxProblemDEQSeriesPageEmpty4}{\showpaste}
\tab{5}\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y }\free{eq }\free{c7 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPagePatch5}
\begin{paste}{ugxProblemDEQSeriesPageFull5}{ugxProblemDEQSeriesPageEmpty5}
\pastebutton{ugxProblemDEQSeriesPageFull5}{\hidepaste}
\tab{5}\spadcommand{x := operator 'x\bound{x }}
\indentrel{3}\begin{verbatim}
   (4)  x
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPageEmpty5}
\begin{paste}{ugxProblemDEQSeriesPageEmpty5}{ugxProblemDEQSeriesPagePatch5}
\pastebutton{ugxProblemDEQSeriesPageEmpty5}{\showpaste}
\tab{5}\spadcommand{x := operator 'x\bound{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPagePatch6}
\begin{paste}{ugxProblemDEQSeriesPageFull6}{ugxProblemDEQSeriesPageEmpty6}
\pastebutton{ugxProblemDEQSeriesPageFull6}{\hidepaste}
\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }}
\indentrel{3}\begin{verbatim}
         ,         2
   (5)  x (t)= x(t)  + 1

                      Type: Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPageEmpty6}
\begin{paste}{ugxProblemDEQSeriesPageEmpty6}{ugxProblemDEQSeriesPagePatch6}
\pastebutton{ugxProblemDEQSeriesPageEmpty6}{\showpaste}
\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPagePatch7}
\begin{paste}{ugxProblemDEQSeriesPageFull7}{ugxProblemDEQSeriesPageEmpty7}
\pastebutton{ugxProblemDEQSeriesPageFull7}{\hidepaste}
\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }}
\indentrel{3}\begin{verbatim}
         ,
   (6)  y (t)= x(t)y(t)

                      Type: Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPageEmpty7}
\begin{paste}{ugxProblemDEQSeriesPageEmpty7}{ugxProblemDEQSeriesPagePatch7}
\pastebutton{ugxProblemDEQSeriesPageEmpty7}{\showpaste}
\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPagePatch8}
\begin{paste}{ugxProblemDEQSeriesPageFull8}{ugxProblemDEQSeriesPageEmpty8}
\pastebutton{ugxProblemDEQSeriesPageFull8}{\hidepaste}
\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }\free{c7 }}
\indentrel{3}\begin{verbatim}
   (7)
        1  3    2  5    17  7      8
   [t + Ä t  + ÄÄ t  + ÄÄÄ t  + O(t ),
        3      15      315
        1  2    5  4    61  6      8
    1 + Ä t  + ÄÄ t  + ÄÄÄ t  + O(t )]
        2      24      720
Type: List UnivariateTaylorSeries(Expression Integer,t,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemDEQSeriesPageEmpty8}
\begin{paste}{ugxProblemDEQSeriesPageEmpty8}{ugxProblemDEQSeriesPagePatch8}
\pastebutton{ugxProblemDEQSeriesPageEmpty8}{\showpaste}
\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }\free{c7 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPagePatch1}
\begin{paste}{ugxProblemSeriesCoefficientsPageFull1}{ugxProblemSeriesCoefficientsPageEmpty1}
\pastebutton{ugxProblemSeriesCoefficientsPageFull1}{\hidepaste}
\tab{5}\spadcommand{x := series(x)\bound{x }}
\indentrel{3}\begin{verbatim}
   (1)  x
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty1}
\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty1}{ugxProblemSeriesCoefficientsPagePatch1}
\pastebutton{ugxProblemSeriesCoefficientsPageEmpty1}{\showpaste}
\tab{5}\spadcommand{x := series(x)\bound{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPagePatch2}
\begin{paste}{ugxProblemSeriesCoefficientsPageFull2}{ugxProblemSeriesCoefficientsPageEmpty2}
\pastebutton{ugxProblemSeriesCoefficientsPageFull2}{\hidepaste}
\tab{5}\spadcommand{y := exp(x) * sin(x)\free{x }\bound{y }}
\indentrel{3}\begin{verbatim}
   (2)
          2   1  3    1  5    1  6    1   7     1    9
     x + x  + Ä x  - ÄÄ x  - ÄÄ x  - ÄÄÄ x  + ÄÄÄÄÄ x
              3      30      90      630      22680
   + 
        1    10      1     11      12
     ÄÄÄÄÄÄ x   + ÄÄÄÄÄÄÄ x   + O(x  )
     113400       1247400
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty2}
\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty2}{ugxProblemSeriesCoefficientsPagePatch2}
\pastebutton{ugxProblemSeriesCoefficientsPageEmpty2}{\showpaste}
\tab{5}\spadcommand{y := exp(x) * sin(x)\free{x }\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPagePatch3}
\begin{paste}{ugxProblemSeriesCoefficientsPageFull3}{ugxProblemSeriesCoefficientsPageEmpty3}
\pastebutton{ugxProblemSeriesCoefficientsPageFull3}{\hidepaste}
\tab{5}\spadcommand{coefficient(y,6)\free{y }}
\indentrel{3}\begin{verbatim}
           1
   (3)  - ÄÄ
          90
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty3}
\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty3}{ugxProblemSeriesCoefficientsPagePatch3}
\pastebutton{ugxProblemSeriesCoefficientsPageEmpty3}{\showpaste}
\tab{5}\spadcommand{coefficient(y,6)\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPagePatch4}
\begin{paste}{ugxProblemSeriesCoefficientsPageFull4}{ugxProblemSeriesCoefficientsPageEmpty4}
\pastebutton{ugxProblemSeriesCoefficientsPageFull4}{\hidepaste}
\tab{5}\spadcommand{coefficient(y,15)\free{y }\bound{y15 }}
\indentrel{3}\begin{verbatim}
               1
   (4)  - ÄÄÄÄÄÄÄÄÄÄÄ
          10216206000
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty4}
\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty4}{ugxProblemSeriesCoefficientsPagePatch4}
\pastebutton{ugxProblemSeriesCoefficientsPageEmpty4}{\showpaste}
\tab{5}\spadcommand{coefficient(y,15)\free{y }\bound{y15 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPagePatch5}
\begin{paste}{ugxProblemSeriesCoefficientsPageFull5}{ugxProblemSeriesCoefficientsPageEmpty5}
\pastebutton{ugxProblemSeriesCoefficientsPageFull5}{\hidepaste}
\tab{5}\spadcommand{y\free{y15 }}
\indentrel{3}\begin{verbatim}
   (5)
          2   1  3    1  5    1  6    1   7     1    9
     x + x  + Ä x  - ÄÄ x  - ÄÄ x  - ÄÄÄ x  + ÄÄÄÄÄ x
              3      30      90      630      22680
   + 
        1    10      1     11       1     13
     ÄÄÄÄÄÄ x   + ÄÄÄÄÄÄÄ x   - ÄÄÄÄÄÄÄÄ x
     113400       1247400       97297200
   + 
           1      14        1       15      16
     - ÄÄÄÄÄÄÄÄÄ x   - ÄÄÄÄÄÄÄÄÄÄÄ x   + O(x  )
       681080400       10216206000
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty5}
\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty5}{ugxProblemSeriesCoefficientsPagePatch5}
\pastebutton{ugxProblemSeriesCoefficientsPageEmpty5}{\showpaste}
\tab{5}\spadcommand{y\free{y15 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch1}
\begin{paste}{ugxProblemLDEQClosedPageFull1}{ugxProblemLDEQClosedPageEmpty1}
\pastebutton{ugxProblemLDEQClosedPageFull1}{\hidepaste}
\tab{5}\spadcommand{y := operator 'y\bound{y }}
\indentrel{3}\begin{verbatim}
   (1)  y
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty1}
\begin{paste}{ugxProblemLDEQClosedPageEmpty1}{ugxProblemLDEQClosedPagePatch1}
\pastebutton{ugxProblemLDEQClosedPageEmpty1}{\showpaste}
\tab{5}\spadcommand{y := operator 'y\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch2}
\begin{paste}{ugxProblemLDEQClosedPageFull2}{ugxProblemLDEQClosedPageEmpty2}
\pastebutton{ugxProblemLDEQClosedPageFull2}{\hidepaste}
\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }}
\indentrel{3}\begin{verbatim}
         ,,       ,
   (2)  y  (x) + y (x) + y(x)= 0

                      Type: Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty2}
\begin{paste}{ugxProblemLDEQClosedPageEmpty2}{ugxProblemLDEQClosedPagePatch2}
\pastebutton{ugxProblemLDEQClosedPageEmpty2}{\showpaste}
\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch3}
\begin{paste}{ugxProblemLDEQClosedPageFull3}{ugxProblemLDEQClosedPageEmpty3}
\pastebutton{ugxProblemLDEQClosedPageFull3}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }}
\indentrel{3}\begin{verbatim}
   (3)
   [particular= 0,
                          x     x
                  ÚÄ¿   - Ä   - Ä      ÚÄ¿
                x\³3      2     2    x\³3
    basis= [cos(ÄÄÄÄÄ)%e   ,%e   sin(ÄÄÄÄÄ)]]
                  2                    2
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty3}
\begin{paste}{ugxProblemLDEQClosedPageEmpty3}{ugxProblemLDEQClosedPagePatch3}
\pastebutton{ugxProblemLDEQClosedPageEmpty3}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch4}
\begin{paste}{ugxProblemLDEQClosedPageFull4}{ugxProblemLDEQClosedPageEmpty4}
\pastebutton{ugxProblemLDEQClosedPageFull4}{\hidepaste}
\tab{5}\spadcommand{deq := D(y x, x, 2) + y x\bound{e2 }\free{y }}
\indentrel{3}\begin{verbatim}
         ,,
   (4)  y  (x) + y(x)

                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty4}
\begin{paste}{ugxProblemLDEQClosedPageEmpty4}{ugxProblemLDEQClosedPagePatch4}
\pastebutton{ugxProblemLDEQClosedPageEmpty4}{\showpaste}
\tab{5}\spadcommand{deq := D(y x, x, 2) + y x\bound{e2 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch5}
\begin{paste}{ugxProblemLDEQClosedPageFull5}{ugxProblemLDEQClosedPageEmpty5}
\pastebutton{ugxProblemLDEQClosedPageFull5}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x = 0, [1, 1])\free{e2 }\free{y }}
\indentrel{3}\begin{verbatim}
   (5)  sin(x) + cos(x)
                    Type: Union(Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty5}
\begin{paste}{ugxProblemLDEQClosedPageEmpty5}{ugxProblemLDEQClosedPagePatch5}
\pastebutton{ugxProblemLDEQClosedPageEmpty5}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x = 0, [1, 1])\free{e2 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch6}
\begin{paste}{ugxProblemLDEQClosedPageFull6}{ugxProblemLDEQClosedPageEmpty6}
\pastebutton{ugxProblemLDEQClosedPageFull6}{\hidepaste}
\tab{5}\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4\bound{e3 }\free{y }}
\indentrel{3}\begin{verbatim}
         3 ,,,       2 ,,         ,               4
   (6)  x y   (x) + x y  (x) - 2xy (x) + 2y(x)= 2x

                      Type: Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty6}
\begin{paste}{ugxProblemLDEQClosedPageEmpty6}{ugxProblemLDEQClosedPagePatch6}
\pastebutton{ugxProblemLDEQClosedPageEmpty6}{\showpaste}
\tab{5}\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4\bound{e3 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch7}
\begin{paste}{ugxProblemLDEQClosedPageFull7}{ugxProblemLDEQClosedPageEmpty7}
\pastebutton{ugxProblemLDEQClosedPageFull7}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }}
\indentrel{3}\begin{verbatim}
   (7)
                 5      3      2
                x  - 10x  + 20x  + 4
   [particular= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
                         15x
              3     2      3      3     2
            2x  - 3x  + 1 x  - 1 x  - 3x  - 1
    basis= [ÄÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄ]]
                  x          x         x
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty7}
\begin{paste}{ugxProblemLDEQClosedPageEmpty7}{ugxProblemLDEQClosedPagePatch7}
\pastebutton{ugxProblemLDEQClosedPageEmpty7}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch8}
\begin{paste}{ugxProblemLDEQClosedPageFull8}{ugxProblemLDEQClosedPageEmpty8}
\pastebutton{ugxProblemLDEQClosedPageFull8}{\hidepaste}
\tab{5}\spadcommand{deq := (x**9+x**3) * D(y x, x, 3) + 18 * x**8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x**6 - 3) * y x\bound{e4 }\free{y }}
\indentrel{3}\begin{verbatim}
   (8)
       9    3  ,,,         8 ,,          ,
     (x  + x )y   (x) + 18x y  (x) - 90xy (x)

   + 
            6
     (- 330x  + 90)y(x)
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty8}
\begin{paste}{ugxProblemLDEQClosedPageEmpty8}{ugxProblemLDEQClosedPagePatch8}
\pastebutton{ugxProblemLDEQClosedPageEmpty8}{\showpaste}
\tab{5}\spadcommand{deq := (x**9+x**3) * D(y x, x, 3) + 18 * x**8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x**6 - 3) * y x\bound{e4 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch9}
\begin{paste}{ugxProblemLDEQClosedPageFull9}{ugxProblemLDEQClosedPageEmpty9}
\pastebutton{ugxProblemLDEQClosedPageFull9}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e4 }\free{y }}
\indentrel{3}\begin{verbatim}
   (9)
   [particular= 0,
                          ÚÄÄ¿            ÚÄÄ¿
                       - \³91 log(x)     \³91 log(x)
               x   x %e              x %e
    basis= [ÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]]
             6            6                6
            x  + 1       x  + 1           x  + 1
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty9}
\begin{paste}{ugxProblemLDEQClosedPageEmpty9}{ugxProblemLDEQClosedPagePatch9}
\pastebutton{ugxProblemLDEQClosedPageEmpty9}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e4 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch10}
\begin{paste}{ugxProblemLDEQClosedPageFull10}{ugxProblemLDEQClosedPageEmpty10}
\pastebutton{ugxProblemLDEQClosedPageFull10}{\hidepaste}
\tab{5}\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }}
\indentrel{3}\begin{verbatim}
           2      ,,         ,
   (10)  (x  + 1)y  (x) + 3xy (x) + y(x)= 0

                      Type: Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty10}
\begin{paste}{ugxProblemLDEQClosedPageEmpty10}{ugxProblemLDEQClosedPagePatch10}
\pastebutton{ugxProblemLDEQClosedPageEmpty10}{\showpaste}
\tab{5}\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPagePatch11}
\begin{paste}{ugxProblemLDEQClosedPageFull11}{ugxProblemLDEQClosedPageEmpty11}
\pastebutton{ugxProblemLDEQClosedPageFull11}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }}
\indentrel{3}\begin{verbatim}
   (11)
                                         ÚÄÄÄÄÄÄ¿
                                         ³ 2
                              1     log(\³x  + 1  - x)
   [particular= 0,basis= [ÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]]
                           ÚÄÄÄÄÄÄ¿       ÚÄÄÄÄÄÄ¿
                           ³ 2            ³ 2
                          \³x  + 1       \³x  + 1
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemLDEQClosedPageEmpty11}
\begin{paste}{ugxProblemLDEQClosedPageEmpty11}{ugxProblemLDEQClosedPagePatch11}
\pastebutton{ugxProblemLDEQClosedPageEmpty11}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch1}
\begin{paste}{ugxProblemNLDEQClosedPageFull1}{ugxProblemNLDEQClosedPageEmpty1}
\pastebutton{ugxProblemNLDEQClosedPageFull1}{\hidepaste}
\tab{5}\spadcommand{m := -y\bound{m }}
\indentrel{3}\begin{verbatim}
   (1)  - y
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty1}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty1}{ugxProblemNLDEQClosedPagePatch1}
\pastebutton{ugxProblemNLDEQClosedPageEmpty1}{\showpaste}
\tab{5}\spadcommand{m := -y\bound{m }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch2}
\begin{paste}{ugxProblemNLDEQClosedPageFull2}{ugxProblemNLDEQClosedPageEmpty2}
\pastebutton{ugxProblemNLDEQClosedPageFull2}{\hidepaste}
\tab{5}\spadcommand{n := x + y * log y\bound{n }}
\indentrel{3}\begin{verbatim}
   (2)  y log(y) + x
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty2}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty2}{ugxProblemNLDEQClosedPagePatch2}
\pastebutton{ugxProblemNLDEQClosedPageEmpty2}{\showpaste}
\tab{5}\spadcommand{n := x + y * log y\bound{n }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch3}
\begin{paste}{ugxProblemNLDEQClosedPageFull3}{ugxProblemNLDEQClosedPageEmpty3}
\pastebutton{ugxProblemNLDEQClosedPageFull3}{\hidepaste}
\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m n }}
\indentrel{3}\begin{verbatim}
   (3)  - 2
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty3}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty3}{ugxProblemNLDEQClosedPagePatch3}
\pastebutton{ugxProblemNLDEQClosedPageEmpty3}{\showpaste}
\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m n }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch4}
\begin{paste}{ugxProblemNLDEQClosedPageFull4}{ugxProblemNLDEQClosedPageEmpty4}
\pastebutton{ugxProblemNLDEQClosedPageFull4}{\hidepaste}
\tab{5}\spadcommand{mu := operator 'mu\bound{mu }}
\indentrel{3}\begin{verbatim}
   (4)  mu
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty4}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty4}{ugxProblemNLDEQClosedPagePatch4}
\pastebutton{ugxProblemNLDEQClosedPageEmpty4}{\showpaste}
\tab{5}\spadcommand{mu := operator 'mu\bound{mu }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch5}
\begin{paste}{ugxProblemNLDEQClosedPageFull5}{ugxProblemNLDEQClosedPageEmpty5}
\pastebutton{ugxProblemNLDEQClosedPageFull5}{\hidepaste}
\tab{5}\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x)\bound{a }\free{m n mu }}
\indentrel{3}\begin{verbatim}
                          ,
   (5)  (- y log(y) - x)mu (x) - 2mu(x)

                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty5}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty5}{ugxProblemNLDEQClosedPagePatch5}
\pastebutton{ugxProblemNLDEQClosedPageEmpty5}{\showpaste}
\tab{5}\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x)\bound{a }\free{m n mu }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch6}
\begin{paste}{ugxProblemNLDEQClosedPageFull6}{ugxProblemNLDEQClosedPageEmpty6}
\pastebutton{ugxProblemNLDEQClosedPageFull6}{\hidepaste}
\tab{5}\spadcommand{solve(a = 0, mu, x)\free{mu a }}
\indentrel{3}\begin{verbatim}
   (6)
                                        1
   [particular= 0,basis= [ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]]
                           2      2                  2
                          y log(y)  + 2x y log(y) + x
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty6}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty6}{ugxProblemNLDEQClosedPagePatch6}
\pastebutton{ugxProblemNLDEQClosedPageEmpty6}{\showpaste}
\tab{5}\spadcommand{solve(a = 0, mu, x)\free{mu a }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch7}
\begin{paste}{ugxProblemNLDEQClosedPageFull7}{ugxProblemNLDEQClosedPageEmpty7}
\pastebutton{ugxProblemNLDEQClosedPageFull7}{\hidepaste}
\tab{5}\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x)\bound{b }\free{mu m }}
\indentrel{3}\begin{verbatim}
             ,
   (7)  - ymu (y) - 2mu(y)

                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty7}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty7}{ugxProblemNLDEQClosedPagePatch7}
\pastebutton{ugxProblemNLDEQClosedPageEmpty7}{\showpaste}
\tab{5}\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x)\bound{b }\free{mu m }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch8}
\begin{paste}{ugxProblemNLDEQClosedPageFull8}{ugxProblemNLDEQClosedPageEmpty8}
\pastebutton{ugxProblemNLDEQClosedPageFull8}{\hidepaste}
\tab{5}\spadcommand{sb := solve(b = 0, mu, y)\free{mu b }\bound{sb }}
\indentrel{3}\begin{verbatim}
                                1
   (8)  [particular= 0,basis= [ÄÄ]]
                                2
                               y
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty8}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty8}{ugxProblemNLDEQClosedPagePatch8}
\pastebutton{ugxProblemNLDEQClosedPageEmpty8}{\showpaste}
\tab{5}\spadcommand{sb := solve(b = 0, mu, y)\free{mu b }\bound{sb }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch9}
\begin{paste}{ugxProblemNLDEQClosedPageFull9}{ugxProblemNLDEQClosedPageEmpty9}
\pastebutton{ugxProblemNLDEQClosedPageFull9}{\hidepaste}
\tab{5}\spadcommand{intFactor := sb.basis.1\bound{intFactor }\free{sb }}
\indentrel{3}\begin{verbatim}
         1
   (9)  ÄÄ
         2
        y
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty9}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty9}{ugxProblemNLDEQClosedPagePatch9}
\pastebutton{ugxProblemNLDEQClosedPageEmpty9}{\showpaste}
\tab{5}\spadcommand{intFactor := sb.basis.1\bound{intFactor }\free{sb }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch10}
\begin{paste}{ugxProblemNLDEQClosedPageFull10}{ugxProblemNLDEQClosedPageEmpty10}
\pastebutton{ugxProblemNLDEQClosedPageFull10}{\hidepaste}
\tab{5}\spadcommand{m := intFactor * m\bound{m1 }\free{m intFactor }}
\indentrel{3}\begin{verbatim}
           1
   (10)  - Ä
           y
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty10}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty10}{ugxProblemNLDEQClosedPagePatch10}
\pastebutton{ugxProblemNLDEQClosedPageEmpty10}{\showpaste}
\tab{5}\spadcommand{m := intFactor * m\bound{m1 }\free{m intFactor }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch11}
\begin{paste}{ugxProblemNLDEQClosedPageFull11}{ugxProblemNLDEQClosedPageEmpty11}
\pastebutton{ugxProblemNLDEQClosedPageFull11}{\hidepaste}
\tab{5}\spadcommand{n := intFactor * n\bound{n1 }\free{n intFactor }}
\indentrel{3}\begin{verbatim}
         y log(y) + x
   (11)  ÄÄÄÄÄÄÄÄÄÄÄÄ
               2
              y
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty11}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty11}{ugxProblemNLDEQClosedPagePatch11}
\pastebutton{ugxProblemNLDEQClosedPageEmpty11}{\showpaste}
\tab{5}\spadcommand{n := intFactor * n\bound{n1 }\free{n intFactor }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch12}
\begin{paste}{ugxProblemNLDEQClosedPageFull12}{ugxProblemNLDEQClosedPageEmpty12}
\pastebutton{ugxProblemNLDEQClosedPageFull12}{\hidepaste}
\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m1 n1 }}
\indentrel{3}\begin{verbatim}
   (12)  0
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty12}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty12}{ugxProblemNLDEQClosedPagePatch12}
\pastebutton{ugxProblemNLDEQClosedPageEmpty12}{\showpaste}
\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m1 n1 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch13}
\begin{paste}{ugxProblemNLDEQClosedPageFull13}{ugxProblemNLDEQClosedPageEmpty13}
\pastebutton{ugxProblemNLDEQClosedPageFull13}{\hidepaste}
\tab{5}\spadcommand{h := operator 'h\bound{h }}
\indentrel{3}\begin{verbatim}
   (13)  h
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty13}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty13}{ugxProblemNLDEQClosedPagePatch13}
\pastebutton{ugxProblemNLDEQClosedPageEmpty13}{\showpaste}
\tab{5}\spadcommand{h := operator 'h\bound{h }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch14}
\begin{paste}{ugxProblemNLDEQClosedPageFull14}{ugxProblemNLDEQClosedPageEmpty14}
\pastebutton{ugxProblemNLDEQClosedPageFull14}{\hidepaste}
\tab{5}\spadcommand{sol := h y + integrate(m, x)\bound{sol }\free{h m1 }}
\indentrel{3}\begin{verbatim}
         y h(y) - x
   (14)  ÄÄÄÄÄÄÄÄÄÄ
              y
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty14}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty14}{ugxProblemNLDEQClosedPagePatch14}
\pastebutton{ugxProblemNLDEQClosedPageEmpty14}{\showpaste}
\tab{5}\spadcommand{sol := h y + integrate(m, x)\bound{sol }\free{h m1 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch15}
\begin{paste}{ugxProblemNLDEQClosedPageFull15}{ugxProblemNLDEQClosedPageEmpty15}
\pastebutton{ugxProblemNLDEQClosedPageFull15}{\hidepaste}
\tab{5}\spadcommand{dsol := D(sol, y)\free{sol }\bound{dsol }}
\indentrel{3}\begin{verbatim}
          2 ,
         y h (y) + x

   (15)  ÄÄÄÄÄÄÄÄÄÄÄ
               2
              y
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty15}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty15}{ugxProblemNLDEQClosedPagePatch15}
\pastebutton{ugxProblemNLDEQClosedPageEmpty15}{\showpaste}
\tab{5}\spadcommand{dsol := D(sol, y)\free{sol }\bound{dsol }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch16}
\begin{paste}{ugxProblemNLDEQClosedPageFull16}{ugxProblemNLDEQClosedPageEmpty16}
\pastebutton{ugxProblemNLDEQClosedPageFull16}{\hidepaste}
\tab{5}\spadcommand{nsol := solve(dsol = n, h, y)\free{dsol n1 h }\bound{nsol }}
\indentrel{3}\begin{verbatim}
                            2
                      log(y)
   (16)  [particular= ÄÄÄÄÄÄÄ,basis= [1]]
                         2
Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty16}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty16}{ugxProblemNLDEQClosedPagePatch16}
\pastebutton{ugxProblemNLDEQClosedPageEmpty16}{\showpaste}
\tab{5}\spadcommand{nsol := solve(dsol = n, h, y)\free{dsol n1 h }\bound{nsol }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch17}
\begin{paste}{ugxProblemNLDEQClosedPageFull17}{ugxProblemNLDEQClosedPageEmpty17}
\pastebutton{ugxProblemNLDEQClosedPageFull17}{\hidepaste}
\tab{5}\spadcommand{eval(sol, h y = nsol.particular)\free{sol h nsol }}
\indentrel{3}\begin{verbatim}
                 2
         y log(y)  - 2x
   (17)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
               2y
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty17}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty17}{ugxProblemNLDEQClosedPagePatch17}
\pastebutton{ugxProblemNLDEQClosedPageEmpty17}{\showpaste}
\tab{5}\spadcommand{eval(sol, h y = nsol.particular)\free{sol h nsol }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch18}
\begin{paste}{ugxProblemNLDEQClosedPageFull18}{ugxProblemNLDEQClosedPageEmpty18}
\pastebutton{ugxProblemNLDEQClosedPageFull18}{\hidepaste}
\tab{5}\spadcommand{y := operator 'y\bound{y }}
\indentrel{3}\begin{verbatim}
   (18)  y
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty18}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty18}{ugxProblemNLDEQClosedPagePatch18}
\pastebutton{ugxProblemNLDEQClosedPageEmpty18}{\showpaste}
\tab{5}\spadcommand{y := operator 'y\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch19}
\begin{paste}{ugxProblemNLDEQClosedPageFull19}{ugxProblemNLDEQClosedPageEmpty19}
\pastebutton{ugxProblemNLDEQClosedPageFull19}{\hidepaste}
\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }}
\indentrel{3}\begin{verbatim}
          ,            y(x)
   (19)  y (x)= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                y(x)log(y(x)) + x
                      Type: Equation Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty19}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty19}{ugxProblemNLDEQClosedPagePatch19}
\pastebutton{ugxProblemNLDEQClosedPageEmpty19}{\showpaste}
\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPagePatch20}
\begin{paste}{ugxProblemNLDEQClosedPageFull20}{ugxProblemNLDEQClosedPageEmpty20}
\pastebutton{ugxProblemNLDEQClosedPageFull20}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }}
\indentrel{3}\begin{verbatim}
                      2
         y(x)log(y(x))  - 2x
   (20)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                2y(x)
                    Type: Union(Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemNLDEQClosedPageEmpty20}
\begin{paste}{ugxProblemNLDEQClosedPageEmpty20}{ugxProblemNLDEQClosedPagePatch20}
\pastebutton{ugxProblemNLDEQClosedPageEmpty20}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch1}
\begin{paste}{ugxProblemSymRootAllPageFull1}{ugxProblemSymRootAllPageEmpty1}
\pastebutton{ugxProblemSymRootAllPageFull1}{\hidepaste}
\tab{5}\spadcommand{l := rootsOf(x**4+1,x)\bound{l }}
\indentrel{3}\begin{verbatim}
   (1)  [%x0,%x0 %x1,- %x0,- %x0 %x1]
                          Type: List Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty1}
\begin{paste}{ugxProblemSymRootAllPageEmpty1}{ugxProblemSymRootAllPagePatch1}
\pastebutton{ugxProblemSymRootAllPageEmpty1}{\showpaste}
\tab{5}\spadcommand{l := rootsOf(x**4+1,x)\bound{l }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch2}
\begin{paste}{ugxProblemSymRootAllPageFull2}{ugxProblemSymRootAllPageEmpty2}
\pastebutton{ugxProblemSymRootAllPageFull2}{\hidepaste}
\tab{5}\spadcommand{\%x0**5\free{l }}
\indentrel{3}\begin{verbatim}
   (2)  - %x0
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty2}
\begin{paste}{ugxProblemSymRootAllPageEmpty2}{ugxProblemSymRootAllPagePatch2}
\pastebutton{ugxProblemSymRootAllPageEmpty2}{\showpaste}
\tab{5}\spadcommand{\%x0**5\free{l }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch3}
\begin{paste}{ugxProblemSymRootAllPageFull3}{ugxProblemSymRootAllPageEmpty3}
\pastebutton{ugxProblemSymRootAllPageFull3}{\hidepaste}
\tab{5}\spadcommand{definingPolynomial \%x0\free{l }}
\indentrel{3}\begin{verbatim}
           4
   (3)  %x0  + 1
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty3}
\begin{paste}{ugxProblemSymRootAllPageEmpty3}{ugxProblemSymRootAllPagePatch3}
\pastebutton{ugxProblemSymRootAllPageEmpty3}{\showpaste}
\tab{5}\spadcommand{definingPolynomial \%x0\free{l }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch4}
\begin{paste}{ugxProblemSymRootAllPageFull4}{ugxProblemSymRootAllPageEmpty4}
\pastebutton{ugxProblemSymRootAllPageFull4}{\hidepaste}
\tab{5}\spadcommand{definingPolynomial \%x1\free{l }}
\indentrel{3}\begin{verbatim}
           2
   (4)  %x1  + 1
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty4}
\begin{paste}{ugxProblemSymRootAllPageEmpty4}{ugxProblemSymRootAllPagePatch4}
\pastebutton{ugxProblemSymRootAllPageEmpty4}{\showpaste}
\tab{5}\spadcommand{definingPolynomial \%x1\free{l }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch5}
\begin{paste}{ugxProblemSymRootAllPageFull5}{ugxProblemSymRootAllPageEmpty5}
\pastebutton{ugxProblemSymRootAllPageFull5}{\hidepaste}
\tab{5}\spadcommand{definingPolynomial \%x2\free{l }}
\indentrel{3}\begin{verbatim}
   (5)  - %x2 + %%var
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty5}
\begin{paste}{ugxProblemSymRootAllPageEmpty5}{ugxProblemSymRootAllPagePatch5}
\pastebutton{ugxProblemSymRootAllPageEmpty5}{\showpaste}
\tab{5}\spadcommand{definingPolynomial \%x2\free{l }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch6}
\begin{paste}{ugxProblemSymRootAllPageFull6}{ugxProblemSymRootAllPageEmpty6}
\pastebutton{ugxProblemSymRootAllPageFull6}{\hidepaste}
\tab{5}\spadcommand{x3 := last l\free{l }\bound{x3 }}
\indentrel{3}\begin{verbatim}
   (6)  - %x0 %x1
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty6}
\begin{paste}{ugxProblemSymRootAllPageEmpty6}{ugxProblemSymRootAllPagePatch6}
\pastebutton{ugxProblemSymRootAllPageEmpty6}{\showpaste}
\tab{5}\spadcommand{x3 := last l\free{l }\bound{x3 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch7}
\begin{paste}{ugxProblemSymRootAllPageFull7}{ugxProblemSymRootAllPageEmpty7}
\pastebutton{ugxProblemSymRootAllPageFull7}{\hidepaste}
\tab{5}\spadcommand{\%x0 + \%x1 + \%x2 + x3\free{x3 }}
\indentrel{3}\begin{verbatim}
   (7)  (- %x0 + 1)%x1 + %x0 + %x2
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty7}
\begin{paste}{ugxProblemSymRootAllPageEmpty7}{ugxProblemSymRootAllPagePatch7}
\pastebutton{ugxProblemSymRootAllPageEmpty7}{\showpaste}
\tab{5}\spadcommand{\%x0 + \%x1 + \%x2 + x3\free{x3 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch8}
\begin{paste}{ugxProblemSymRootAllPageFull8}{ugxProblemSymRootAllPageEmpty8}
\pastebutton{ugxProblemSymRootAllPageFull8}{\hidepaste}
\tab{5}\spadcommand{\%x0 * \%x1 * \%x2 * x3\free{x3 }}
\indentrel{3}\begin{verbatim}
               2
   (8)  %x2 %x0
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty8}
\begin{paste}{ugxProblemSymRootAllPageEmpty8}{ugxProblemSymRootAllPagePatch8}
\pastebutton{ugxProblemSymRootAllPageEmpty8}{\showpaste}
\tab{5}\spadcommand{\%x0 * \%x1 * \%x2 * x3\free{x3 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch9}
\begin{paste}{ugxProblemSymRootAllPageFull9}{ugxProblemSymRootAllPageEmpty9}
\pastebutton{ugxProblemSymRootAllPageFull9}{\hidepaste}
\tab{5}\spadcommand{zerosOf(y**4+1,y)\bound{z }}
\indentrel{3}\begin{verbatim}
   (9)
     ÚÄÄÄ¿      ÚÄÄÄ¿        ÚÄÄÄ¿        ÚÄÄÄ¿
    \³- 1  + 1 \³- 1  - 1 - \³- 1  - 1 - \³- 1  + 1
   [ÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄ]
        ÚÄ¿        ÚÄ¿         ÚÄ¿          ÚÄ¿
       \³2        \³2         \³2          \³2
                          Type: List Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty9}
\begin{paste}{ugxProblemSymRootAllPageEmpty9}{ugxProblemSymRootAllPagePatch9}
\pastebutton{ugxProblemSymRootAllPageEmpty9}{\showpaste}
\tab{5}\spadcommand{zerosOf(y**4+1,y)\bound{z }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPagePatch10}
\begin{paste}{ugxProblemSymRootAllPageFull10}{ugxProblemSymRootAllPageEmpty10}
\pastebutton{ugxProblemSymRootAllPageFull10}{\hidepaste}
\tab{5}\spadcommand{definingPolynomial \%y1\free{z }}
\indentrel{3}\begin{verbatim}
              2
   (10)  %%var  + 1
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootAllPageEmpty10}
\begin{paste}{ugxProblemSymRootAllPageEmpty10}{ugxProblemSymRootAllPagePatch10}
\pastebutton{ugxProblemSymRootAllPageEmpty10}{\showpaste}
\tab{5}\spadcommand{definingPolynomial \%y1\free{z }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch1}
\begin{paste}{ugxProblemSymRootOnePageFull1}{ugxProblemSymRootOnePageEmpty1}
\pastebutton{ugxProblemSymRootOnePageFull1}{\hidepaste}
\tab{5}\spadcommand{a := rootOf(a**4+1,a)\bound{a }}
\indentrel{3}\begin{verbatim}
   (1)  a
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty1}
\begin{paste}{ugxProblemSymRootOnePageEmpty1}{ugxProblemSymRootOnePagePatch1}
\pastebutton{ugxProblemSymRootOnePageEmpty1}{\showpaste}
\tab{5}\spadcommand{a := rootOf(a**4+1,a)\bound{a }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch2}
\begin{paste}{ugxProblemSymRootOnePageFull2}{ugxProblemSymRootOnePageEmpty2}
\pastebutton{ugxProblemSymRootOnePageFull2}{\hidepaste}
\tab{5}\spadcommand{definingPolynomial a\free{a }}
\indentrel{3}\begin{verbatim}
         4
   (2)  a  + 1
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty2}
\begin{paste}{ugxProblemSymRootOnePageEmpty2}{ugxProblemSymRootOnePagePatch2}
\pastebutton{ugxProblemSymRootOnePageEmpty2}{\showpaste}
\tab{5}\spadcommand{definingPolynomial a\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch3}
\begin{paste}{ugxProblemSymRootOnePageFull3}{ugxProblemSymRootOnePageEmpty3}
\pastebutton{ugxProblemSymRootOnePageFull3}{\hidepaste}
\tab{5}\spadcommand{b := rootOf(b**2-a-1,b)\free{a }\bound{b }}
\indentrel{3}\begin{verbatim}
   (3)  b
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty3}
\begin{paste}{ugxProblemSymRootOnePageEmpty3}{ugxProblemSymRootOnePagePatch3}
\pastebutton{ugxProblemSymRootOnePageEmpty3}{\showpaste}
\tab{5}\spadcommand{b := rootOf(b**2-a-1,b)\free{a }\bound{b }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch4}
\begin{paste}{ugxProblemSymRootOnePageFull4}{ugxProblemSymRootOnePageEmpty4}
\pastebutton{ugxProblemSymRootOnePageFull4}{\hidepaste}
\tab{5}\spadcommand{a + b\free{a b }\bound{c }}
\indentrel{3}\begin{verbatim}
   (4)  b + a
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty4}
\begin{paste}{ugxProblemSymRootOnePageEmpty4}{ugxProblemSymRootOnePagePatch4}
\pastebutton{ugxProblemSymRootOnePageEmpty4}{\showpaste}
\tab{5}\spadcommand{a + b\free{a b }\bound{c }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch5}
\begin{paste}{ugxProblemSymRootOnePageFull5}{ugxProblemSymRootOnePageEmpty5}
\pastebutton{ugxProblemSymRootOnePageFull5}{\hidepaste}
\tab{5}\spadcommand{\% ** 5\free{c }}
\indentrel{3}\begin{verbatim}
            3      2                 3      2
   (5)  (10a  + 11a  + 2a - 4)b + 15a  + 10a  + 4a - 10
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty5}
\begin{paste}{ugxProblemSymRootOnePageEmpty5}{ugxProblemSymRootOnePagePatch5}
\pastebutton{ugxProblemSymRootOnePageEmpty5}{\showpaste}
\tab{5}\spadcommand{\% ** 5\free{c }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch6}
\begin{paste}{ugxProblemSymRootOnePageFull6}{ugxProblemSymRootOnePageEmpty6}
\pastebutton{ugxProblemSymRootOnePageFull6}{\hidepaste}
\tab{5}\spadcommand{rootOf(c**2+c+1,c)}
\indentrel{3}\begin{verbatim}
   (6)  c
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty6}
\begin{paste}{ugxProblemSymRootOnePageEmpty6}{ugxProblemSymRootOnePagePatch6}
\pastebutton{ugxProblemSymRootOnePageEmpty6}{\showpaste}
\tab{5}\spadcommand{rootOf(c**2+c+1,c)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch7}
\begin{paste}{ugxProblemSymRootOnePageFull7}{ugxProblemSymRootOnePageEmpty7}
\pastebutton{ugxProblemSymRootOnePageFull7}{\hidepaste}
\tab{5}\spadcommand{zeroOf(d**2+d+1,d)}
\indentrel{3}\begin{verbatim}
         ÚÄÄÄ¿
        \³- 3  - 1
   (7)  ÄÄÄÄÄÄÄÄÄÄ
             2
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty7}
\begin{paste}{ugxProblemSymRootOnePageEmpty7}{ugxProblemSymRootOnePagePatch7}
\pastebutton{ugxProblemSymRootOnePageEmpty7}{\showpaste}
\tab{5}\spadcommand{zeroOf(d**2+d+1,d)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch8}
\begin{paste}{ugxProblemSymRootOnePageFull8}{ugxProblemSymRootOnePageEmpty8}
\pastebutton{ugxProblemSymRootOnePageFull8}{\hidepaste}
\tab{5}\spadcommand{rootOf(e**5-2,e)}
\indentrel{3}\begin{verbatim}
   (8)  e
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty8}
\begin{paste}{ugxProblemSymRootOnePageEmpty8}{ugxProblemSymRootOnePagePatch8}
\pastebutton{ugxProblemSymRootOnePageEmpty8}{\showpaste}
\tab{5}\spadcommand{rootOf(e**5-2,e)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePagePatch9}
\begin{paste}{ugxProblemSymRootOnePageFull9}{ugxProblemSymRootOnePageEmpty9}
\pastebutton{ugxProblemSymRootOnePageFull9}{\hidepaste}
\tab{5}\spadcommand{zeroOf(f**5-2,f)}
\indentrel{3}\begin{verbatim}
        5ÚÄ¿
   (9)  \³2
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSymRootOnePageEmpty9}
\begin{paste}{ugxProblemSymRootOnePageEmpty9}{ugxProblemSymRootOnePagePatch9}
\pastebutton{ugxProblemSymRootOnePageEmpty9}{\showpaste}
\tab{5}\spadcommand{zeroOf(f**5-2,f)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch1}
\begin{paste}{ugProblemIntegrationPageFull1}{ugProblemIntegrationPageEmpty1}
\pastebutton{ugProblemIntegrationPageFull1}{\hidepaste}
\tab{5}\spadcommand{integrate(cosh(a*x)*sinh(a*x), x)}
\indentrel{3}\begin{verbatim}
                 2            2
        sinh(a x)  + cosh(a x)
   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                   4a
                    Type: Union(Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty1}
\begin{paste}{ugProblemIntegrationPageEmpty1}{ugProblemIntegrationPagePatch1}
\pastebutton{ugProblemIntegrationPageEmpty1}{\showpaste}
\tab{5}\spadcommand{integrate(cosh(a*x)*sinh(a*x), x)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch2}
\begin{paste}{ugProblemIntegrationPageFull2}{ugProblemIntegrationPageEmpty2}
\pastebutton{ugProblemIntegrationPageFull2}{\hidepaste}
\tab{5}\spadcommand{integrate(log(1 + sqrt(a * x + b)) / x, x)}
\indentrel{3}\begin{verbatim}
           x      ÚÄÄÄÄÄÄÄÄ¿
         Ú¿  log(\³b + %E a  + 1)
   (2)   ³   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ d%E
        ÀÙ            %E
                    Type: Union(Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty2}
\begin{paste}{ugProblemIntegrationPageEmpty2}{ugProblemIntegrationPagePatch2}
\pastebutton{ugProblemIntegrationPageEmpty2}{\showpaste}
\tab{5}\spadcommand{integrate(log(1 + sqrt(a * x + b)) / x, x)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch3}
\begin{paste}{ugProblemIntegrationPageFull3}{ugProblemIntegrationPageEmpty3}
\pastebutton{ugProblemIntegrationPageFull3}{\hidepaste}
\tab{5}\spadcommand{integrate(1/(x**2 - 2),x)}
\indentrel{3}\begin{verbatim}
              2      ÚÄ¿
            (x  + 2)\³2  - 4x
        log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
                   2
                  x  - 2
   (3)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                   ÚÄ¿
                 2\³2
                    Type: Union(Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty3}
\begin{paste}{ugProblemIntegrationPageEmpty3}{ugProblemIntegrationPagePatch3}
\pastebutton{ugProblemIntegrationPageEmpty3}{\showpaste}
\tab{5}\spadcommand{integrate(1/(x**2 - 2),x)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch4}
\begin{paste}{ugProblemIntegrationPageFull4}{ugProblemIntegrationPageEmpty4}
\pastebutton{ugProblemIntegrationPageFull4}{\hidepaste}
\tab{5}\spadcommand{integrate(1/(x**2 + 2),x)}
\indentrel{3}\begin{verbatim}
               ÚÄ¿
             x\³2
        atan(ÄÄÄÄÄ)
               2
   (4)  ÄÄÄÄÄÄÄÄÄÄÄ
             ÚÄ¿
            \³2
                    Type: Union(Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty4}
\begin{paste}{ugProblemIntegrationPageEmpty4}{ugProblemIntegrationPagePatch4}
\pastebutton{ugProblemIntegrationPageEmpty4}{\showpaste}
\tab{5}\spadcommand{integrate(1/(x**2 + 2),x)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch5}
\begin{paste}{ugProblemIntegrationPageFull5}{ugProblemIntegrationPageEmpty5}
\pastebutton{ugProblemIntegrationPageFull5}{\hidepaste}
\tab{5}\spadcommand{integrate(x**2 / (x**4 - a**2), x)}
\indentrel{3}\begin{verbatim}
   (5)
          2      ÚÄ¿                   ÚÄ¿
        (x  + a)\³a  - 2a x          x\³a
    log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ) + 2atan(ÄÄÄÄÄ)
                2                      a
               x  - a
   [ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
                       ÚÄ¿
                     4\³a
          2      ÚÄÄÄ¿                   ÚÄÄÄ¿
        (x  - a)\³- a  + 2a x          x\³- a
    log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ) - 2atan(ÄÄÄÄÄÄÄ)
                 2                        a
                x  + a
    ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]
                        ÚÄÄÄ¿
                      4\³- a
               Type: Union(List Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty5}
\begin{paste}{ugProblemIntegrationPageEmpty5}{ugProblemIntegrationPagePatch5}
\pastebutton{ugProblemIntegrationPageEmpty5}{\showpaste}
\tab{5}\spadcommand{integrate(x**2 / (x**4 - a**2), x)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch6}
\begin{paste}{ugProblemIntegrationPageFull6}{ugProblemIntegrationPageEmpty6}
\pastebutton{ugProblemIntegrationPageFull6}{\hidepaste}
\tab{5}\spadcommand{complexIntegrate(x**2 / (x**4 - a**2), x)}
\indentrel{3}\begin{verbatim}
   (6)
                  ÚÄÄÄÄ¿                      ÚÄÄ¿
        ÚÄÄ¿    x\³- 4a  + 2a     ÚÄÄÄÄ¿    x\³4a  + 2a
       \³4a log(ÄÄÄÄÄÄÄÄÄÄÄÄÄ) - \³- 4a log(ÄÄÄÄÄÄÄÄÄÄÄ)
                    ÚÄÄÄÄ¿                      ÚÄÄ¿
                   \³- 4a                      \³4a
     + 
                    ÚÄÄ¿                    ÚÄÄÄÄ¿
        ÚÄÄÄÄ¿    x\³4a  - 2a     ÚÄÄ¿    x\³- 4a  - 2a
       \³- 4a log(ÄÄÄÄÄÄÄÄÄÄÄ) - \³4a log(ÄÄÄÄÄÄÄÄÄÄÄÄÄ)
                      ÚÄÄ¿                    ÚÄÄÄÄ¿
                     \³4a                    \³- 4a
  /
       ÚÄÄÄÄ¿ ÚÄÄ¿
     2\³- 4a \³4a
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty6}
\begin{paste}{ugProblemIntegrationPageEmpty6}{ugProblemIntegrationPagePatch6}
\pastebutton{ugProblemIntegrationPageEmpty6}{\showpaste}
\tab{5}\spadcommand{complexIntegrate(x**2 / (x**4 - a**2), x)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch7}
\begin{paste}{ugProblemIntegrationPageFull7}{ugProblemIntegrationPageEmpty7}
\pastebutton{ugProblemIntegrationPageFull7}{\hidepaste}
\tab{5}\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)}
\indentrel{3}\begin{verbatim}
           x      ÚÄÄÄÄÄÄÄÄ¿
         Ú¿  log(\³b + %E a  + 1)
   (7)   ³   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ d%E
        ÀÙ            %E
                               Type: Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty7}
\begin{paste}{ugProblemIntegrationPageEmpty7}{ugProblemIntegrationPagePatch7}
\pastebutton{ugProblemIntegrationPageEmpty7}{\showpaste}
\tab{5}\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch8}
\begin{paste}{ugProblemIntegrationPageFull8}{ugProblemIntegrationPageEmpty8}
\pastebutton{ugProblemIntegrationPageFull8}{\hidepaste}
\tab{5}\spadcommand{integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)}
\indentrel{3}\begin{verbatim}
                                               1
        2atan(8) + 2atan(5) + 2atan(2) + 2atan(Ä) - %pi
                                               2
   (8)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                               2
Type: Union(f1: OrderedCompletion Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty8}
\begin{paste}{ugProblemIntegrationPageEmpty8}{ugProblemIntegrationPagePatch8}
\pastebutton{ugProblemIntegrationPageEmpty8}{\showpaste}
\tab{5}\spadcommand{integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch9}
\begin{paste}{ugProblemIntegrationPageFull9}{ugProblemIntegrationPageEmpty9}
\pastebutton{ugProblemIntegrationPageFull9}{\hidepaste}
\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2)}
\indentrel{3}\begin{verbatim}
   (9)  potentialPole
                   Type: Union(pole: potentialPole,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty9}
\begin{paste}{ugProblemIntegrationPageEmpty9}{ugProblemIntegrationPagePatch9}
\pastebutton{ugProblemIntegrationPageEmpty9}{\showpaste}
\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2)}
\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPagePatch10}
\begin{paste}{ugProblemIntegrationPageFull10}{ugProblemIntegrationPageEmpty10}
\pastebutton{ugProblemIntegrationPageFull10}{\hidepaste}
\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2, "noPole")}
\indentrel{3}\begin{verbatim}
   (10)
   [
                    2       ÚÄ¿    3     2
               (- 4a  - 4a)\³a  + a  + 6a  + a
         - log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
                          2
                         a  - 2a + 1
       + 
                  2        ÚÄ¿    3      2
             (- 8a  - 32a)\³a  + a  + 24a  + 16a
         log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
                          2
                         a  - 8a + 16
    /
         ÚÄ¿
       4\³a
     ,
             ÚÄÄÄ¿          ÚÄÄÄ¿
           2\³- a          \³- a
    - atan(ÄÄÄÄÄÄÄ) + atan(ÄÄÄÄÄÄ)
              a               a
    ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]
                 ÚÄÄÄ¿
                \³- a
Type: Union(f2: List OrderedCompletion Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIntegrationPageEmpty10}
\begin{paste}{ugProblemIntegrationPageEmpty10}{ugProblemIntegrationPagePatch10}
\pastebutton{ugProblemIntegrationPageEmpty10}{\showpaste}
\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2, "noPole")}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorRatFunPagePatch1}
\begin{paste}{ugProblemFactorRatFunPageFull1}{ugProblemFactorRatFunPageEmpty1}
\pastebutton{ugProblemFactorRatFunPageFull1}{\hidepaste}
\tab{5}\spadcommand{factorFraction((x**2-4)/(y**2-4))}
\indentrel{3}\begin{verbatim}
        (x - 2)(x + 2)
   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
        (y - 2)(y + 2)
             Type: Fraction Factored Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorRatFunPageEmpty1}
\begin{paste}{ugProblemFactorRatFunPageEmpty1}{ugProblemFactorRatFunPagePatch1}
\pastebutton{ugProblemFactorRatFunPageEmpty1}{\showpaste}
\tab{5}\spadcommand{factorFraction((x**2-4)/(y**2-4))}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorRatFunPagePatch2}
\begin{paste}{ugProblemFactorRatFunPageFull2}{ugProblemFactorRatFunPageEmpty2}
\pastebutton{ugProblemFactorRatFunPageFull2}{\hidepaste}
\tab{5}\spadcommand{map(factor,(x**2-4)/(y**2-4))}
\indentrel{3}\begin{verbatim}
        (x - 2)(x + 2)
   (2)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
        (y - 2)(y + 2)
             Type: Fraction Factored Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorRatFunPageEmpty2}
\begin{paste}{ugProblemFactorRatFunPageEmpty2}{ugProblemFactorRatFunPagePatch2}
\pastebutton{ugProblemFactorRatFunPageEmpty2}{\showpaste}
\tab{5}\spadcommand{map(factor,(x**2-4)/(y**2-4))}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorIntRatPagePatch1}
\begin{paste}{ugProblemFactorIntRatPageFull1}{ugProblemFactorIntRatPageEmpty1}
\pastebutton{ugProblemFactorIntRatPageFull1}{\hidepaste}
\tab{5}\spadcommand{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12)\bound{v }}
\indentrel{3}\begin{verbatim}
   (1)
         3 3       5       2        6    3        8      5
     - 2x y  + (24x  + 24)y  + (- 4x  - x )y + 48x  + 12x
   + 
        3
     48x  + 12
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorIntRatPageEmpty1}
\begin{paste}{ugProblemFactorIntRatPageEmpty1}{ugProblemFactorIntRatPagePatch1}
\pastebutton{ugProblemFactorIntRatPageEmpty1}{\showpaste}
\tab{5}\spadcommand{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12)\bound{v }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorIntRatPagePatch2}
\begin{paste}{ugProblemFactorIntRatPageFull2}{ugProblemFactorIntRatPageEmpty2}
\pastebutton{ugProblemFactorIntRatPageFull2}{\hidepaste}
\tab{5}\spadcommand{factor v\free{v }}
\indentrel{3}\begin{verbatim}
            3       5         2     3
   (2)  - (x y - 12x  - 12)(2y  + 4x  + 1)
                      Type: Factored Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorIntRatPageEmpty2}
\begin{paste}{ugProblemFactorIntRatPageEmpty2}{ugProblemFactorIntRatPagePatch2}
\pastebutton{ugProblemFactorIntRatPageEmpty2}{\showpaste}
\tab{5}\spadcommand{factor v\free{v }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorIntRatPagePatch3}
\begin{paste}{ugProblemFactorIntRatPageFull3}{ugProblemFactorIntRatPageEmpty3}
\pastebutton{ugProblemFactorIntRatPageFull3}{\hidepaste}
\tab{5}\spadcommand{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12)\bound{w }}
\indentrel{3}\begin{verbatim}
           8     7     6   35  5   95  3     2
   (3)  48x  + 8x  - 2x  + ÄÄ x  + ÄÄ x  + 8x  + 12
                            3       2
                      Type: Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorIntRatPageEmpty3}
\begin{paste}{ugProblemFactorIntRatPageEmpty3}{ugProblemFactorIntRatPagePatch3}
\pastebutton{ugProblemFactorIntRatPageEmpty3}{\showpaste}
\tab{5}\spadcommand{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12)\bound{w }}
\end{paste}\end{patch}

\begin{patch}{ugProblemFactorIntRatPagePatch4}
\begin{paste}{ugProblemFactorIntRatPageFull4}{ugProblemFactorIntRatPageEmpty4}
\pastebutton{ugProblemFactorIntRatPageFull4}{\hidepaste}
\tab{5}\spadcommand{factor w\free{w }}
\indentrel{3}\begin{verbatim}
            3   1  2   1   5    1  3
   (4)  48(x  + Ä x  + Ä)(x  - ÄÄ x  + 1)
                6      4       24
             Type: Factored Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemFactorIntRatPageEmpty4}
\begin{paste}{ugProblemFactorIntRatPageEmpty4}{ugProblemFactorIntRatPagePatch4}
\pastebutton{ugProblemFactorIntRatPageEmpty4}{\showpaste}
\tab{5}\spadcommand{factor w\free{w }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch1}
\begin{paste}{ugxProblemFinitePrimePageFull1}{ugxProblemFinitePrimePageEmpty1}
\pastebutton{ugxProblemFinitePrimePageFull1}{\hidepaste}
\tab{5}\spadcommand{(a,b) : IntegerMod 12\bound{abdec }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty1}
\begin{paste}{ugxProblemFinitePrimePageEmpty1}{ugxProblemFinitePrimePagePatch1}
\pastebutton{ugxProblemFinitePrimePageEmpty1}{\showpaste}
\tab{5}\spadcommand{(a,b) : IntegerMod 12\bound{abdec }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch2}
\begin{paste}{ugxProblemFinitePrimePageFull2}{ugxProblemFinitePrimePageEmpty2}
\pastebutton{ugxProblemFinitePrimePageFull2}{\hidepaste}
\tab{5}\spadcommand{(a, b) := (16, 7)\free{abdec }\bound{a b }}
\indentrel{3}\begin{verbatim}
   (2)  7
                                    Type: IntegerMod 12
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty2}
\begin{paste}{ugxProblemFinitePrimePageEmpty2}{ugxProblemFinitePrimePagePatch2}
\pastebutton{ugxProblemFinitePrimePageEmpty2}{\showpaste}
\tab{5}\spadcommand{(a, b) := (16, 7)\free{abdec }\bound{a b }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch3}
\begin{paste}{ugxProblemFinitePrimePageFull3}{ugxProblemFinitePrimePageEmpty3}
\pastebutton{ugxProblemFinitePrimePageFull3}{\hidepaste}
\tab{5}\spadcommand{[a - b, a * b]\free{a b }}
\indentrel{3}\begin{verbatim}
   (3)  [9,4]
                               Type: List IntegerMod 12
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty3}
\begin{paste}{ugxProblemFinitePrimePageEmpty3}{ugxProblemFinitePrimePagePatch3}
\pastebutton{ugxProblemFinitePrimePageEmpty3}{\showpaste}
\tab{5}\spadcommand{[a - b, a * b]\free{a b }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch4}
\begin{paste}{ugxProblemFinitePrimePageFull4}{ugxProblemFinitePrimePageEmpty4}
\pastebutton{ugxProblemFinitePrimePageFull4}{\hidepaste}
\tab{5}\spadcommand{a / b\free{a b }}
\indentrel{3}\begin{verbatim}
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty4}
\begin{paste}{ugxProblemFinitePrimePageEmpty4}{ugxProblemFinitePrimePagePatch4}
\pastebutton{ugxProblemFinitePrimePageEmpty4}{\showpaste}
\tab{5}\spadcommand{a / b\free{a b }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch5}
\begin{paste}{ugxProblemFinitePrimePageFull5}{ugxProblemFinitePrimePageEmpty5}
\pastebutton{ugxProblemFinitePrimePageFull5}{\hidepaste}
\tab{5}\spadcommand{recip a\free{a }}
\indentrel{3}\begin{verbatim}
   (4)  "failed"
                              Type: Union("failed",...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty5}
\begin{paste}{ugxProblemFinitePrimePageEmpty5}{ugxProblemFinitePrimePagePatch5}
\pastebutton{ugxProblemFinitePrimePageEmpty5}{\showpaste}
\tab{5}\spadcommand{recip a\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch6}
\begin{paste}{ugxProblemFinitePrimePageFull6}{ugxProblemFinitePrimePageEmpty6}
\pastebutton{ugxProblemFinitePrimePageFull6}{\hidepaste}
\tab{5}\spadcommand{recip b\free{b }}
\indentrel{3}\begin{verbatim}
   (5)  7
                         Type: Union(IntegerMod 12,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty6}
\begin{paste}{ugxProblemFinitePrimePageEmpty6}{ugxProblemFinitePrimePagePatch6}
\pastebutton{ugxProblemFinitePrimePageEmpty6}{\showpaste}
\tab{5}\spadcommand{recip b\free{b }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch7}
\begin{paste}{ugxProblemFinitePrimePageFull7}{ugxProblemFinitePrimePageEmpty7}
\pastebutton{ugxProblemFinitePrimePageFull7}{\hidepaste}
\tab{5}\spadcommand{c : PrimeField 11 := 8\bound{c }}
\indentrel{3}\begin{verbatim}
   (6)  8
                                    Type: PrimeField 11
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty7}
\begin{paste}{ugxProblemFinitePrimePageEmpty7}{ugxProblemFinitePrimePagePatch7}
\pastebutton{ugxProblemFinitePrimePageEmpty7}{\showpaste}
\tab{5}\spadcommand{c : PrimeField 11 := 8\bound{c }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch8}
\begin{paste}{ugxProblemFinitePrimePageFull8}{ugxProblemFinitePrimePageEmpty8}
\pastebutton{ugxProblemFinitePrimePageFull8}{\hidepaste}
\tab{5}\spadcommand{inv c\free{c }}
\indentrel{3}\begin{verbatim}
   (7)  7
                                    Type: PrimeField 11
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty8}
\begin{paste}{ugxProblemFinitePrimePageEmpty8}{ugxProblemFinitePrimePagePatch8}
\pastebutton{ugxProblemFinitePrimePageEmpty8}{\showpaste}
\tab{5}\spadcommand{inv c\free{c }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch9}
\begin{paste}{ugxProblemFinitePrimePageFull9}{ugxProblemFinitePrimePageEmpty9}
\pastebutton{ugxProblemFinitePrimePageFull9}{\hidepaste}
\tab{5}\spadcommand{9/c\free{c }}
\indentrel{3}\begin{verbatim}
   (8)  8
                                    Type: PrimeField 11
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty9}
\begin{paste}{ugxProblemFinitePrimePageEmpty9}{ugxProblemFinitePrimePagePatch9}
\pastebutton{ugxProblemFinitePrimePageEmpty9}{\showpaste}
\tab{5}\spadcommand{9/c\free{c }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch10}
\begin{paste}{ugxProblemFinitePrimePageFull10}{ugxProblemFinitePrimePageEmpty10}
\pastebutton{ugxProblemFinitePrimePageFull10}{\hidepaste}
\tab{5}\spadcommand{GF101 := PF 101\bound{GF101 }}
\indentrel{3}\begin{verbatim}
   (9)  PrimeField 101
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty10}
\begin{paste}{ugxProblemFinitePrimePageEmpty10}{ugxProblemFinitePrimePagePatch10}
\pastebutton{ugxProblemFinitePrimePageEmpty10}{\showpaste}
\tab{5}\spadcommand{GF101 := PF 101\bound{GF101 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch11}
\begin{paste}{ugxProblemFinitePrimePageFull11}{ugxProblemFinitePrimePageEmpty11}
\pastebutton{ugxProblemFinitePrimePageFull11}{\hidepaste}
\tab{5}\spadcommand{x := random()$GF101\bound{x }\free{GF101 }}
\indentrel{3}\begin{verbatim}
   (10)  78
                                   Type: PrimeField 101
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty11}
\begin{paste}{ugxProblemFinitePrimePageEmpty11}{ugxProblemFinitePrimePagePatch11}
\pastebutton{ugxProblemFinitePrimePageEmpty11}{\showpaste}
\tab{5}\spadcommand{x := random()$GF101\bound{x }\free{GF101 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch12}
\begin{paste}{ugxProblemFinitePrimePageFull12}{ugxProblemFinitePrimePageEmpty12}
\pastebutton{ugxProblemFinitePrimePageFull12}{\hidepaste}
\tab{5}\spadcommand{y : GF101 := 37\bound{y }\free{GF101 }}
\indentrel{3}\begin{verbatim}
   (11)  37
                                   Type: PrimeField 101
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty12}
\begin{paste}{ugxProblemFinitePrimePageEmpty12}{ugxProblemFinitePrimePagePatch12}
\pastebutton{ugxProblemFinitePrimePageEmpty12}{\showpaste}
\tab{5}\spadcommand{y : GF101 := 37\bound{y }\free{GF101 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch13}
\begin{paste}{ugxProblemFinitePrimePageFull13}{ugxProblemFinitePrimePageEmpty13}
\pastebutton{ugxProblemFinitePrimePageFull13}{\hidepaste}
\tab{5}\spadcommand{z := x/y\bound{z }\free{x y }}
\indentrel{3}\begin{verbatim}
   (12)  84
                                   Type: PrimeField 101
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty13}
\begin{paste}{ugxProblemFinitePrimePageEmpty13}{ugxProblemFinitePrimePagePatch13}
\pastebutton{ugxProblemFinitePrimePageEmpty13}{\showpaste}
\tab{5}\spadcommand{z := x/y\bound{z }\free{x y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch14}
\begin{paste}{ugxProblemFinitePrimePageFull14}{ugxProblemFinitePrimePageEmpty14}
\pastebutton{ugxProblemFinitePrimePageFull14}{\hidepaste}
\tab{5}\spadcommand{z * y - x\free{x y z }}
\indentrel{3}\begin{verbatim}
   (13)  0
                                   Type: PrimeField 101
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty14}
\begin{paste}{ugxProblemFinitePrimePageEmpty14}{ugxProblemFinitePrimePagePatch14}
\pastebutton{ugxProblemFinitePrimePageEmpty14}{\showpaste}
\tab{5}\spadcommand{z * y - x\free{x y z }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch15}
\begin{paste}{ugxProblemFinitePrimePageFull15}{ugxProblemFinitePrimePageEmpty15}
\pastebutton{ugxProblemFinitePrimePageFull15}{\hidepaste}
\tab{5}\spadcommand{pe := primitiveElement()$GF101\bound{pe }\free{GF101 }}
\indentrel{3}\begin{verbatim}
   (14)  2
                                   Type: PrimeField 101
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty15}
\begin{paste}{ugxProblemFinitePrimePageEmpty15}{ugxProblemFinitePrimePagePatch15}
\pastebutton{ugxProblemFinitePrimePageEmpty15}{\showpaste}
\tab{5}\spadcommand{pe := primitiveElement()$GF101\bound{pe }\free{GF101 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch16}
\begin{paste}{ugxProblemFinitePrimePageFull16}{ugxProblemFinitePrimePageEmpty16}
\pastebutton{ugxProblemFinitePrimePageFull16}{\hidepaste}
\tab{5}\spadcommand{[pe**i for i in 0..99]\free{pe }}
\indentrel{3}\begin{verbatim}
   (15)
   [1, 2, 4, 8, 16, 32, 64, 27, 54, 7, 14, 28, 56, 11,
    22, 44, 88, 75, 49, 98, 95, 89, 77, 53, 5, 10, 20,
    40, 80, 59, 17, 34, 68, 35, 70, 39, 78, 55, 9, 18,
    36, 72, 43, 86, 71, 41, 82, 63, 25, 50, 100, 99, 97,
    93, 85, 69, 37, 74, 47, 94, 87, 73, 45, 90, 79, 57,
    13, 26, 52, 3, 6, 12, 24, 48, 96, 91, 81, 61, 21, 42,
    84, 67, 33, 66, 31, 62, 23, 46, 92, 83, 65, 29, 58,
    15, 30, 60, 19, 38, 76, 51]
                              Type: List PrimeField 101
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty16}
\begin{paste}{ugxProblemFinitePrimePageEmpty16}{ugxProblemFinitePrimePagePatch16}
\pastebutton{ugxProblemFinitePrimePageEmpty16}{\showpaste}
\tab{5}\spadcommand{[pe**i for i in 0..99]\free{pe }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch17}
\begin{paste}{ugxProblemFinitePrimePageFull17}{ugxProblemFinitePrimePageEmpty17}
\pastebutton{ugxProblemFinitePrimePageFull17}{\hidepaste}
\tab{5}\spadcommand{ex := discreteLog(y)\bound{ex }\free{y }}
\indentrel{3}\begin{verbatim}
   (16)  56
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty17}
\begin{paste}{ugxProblemFinitePrimePageEmpty17}{ugxProblemFinitePrimePagePatch17}
\pastebutton{ugxProblemFinitePrimePageEmpty17}{\showpaste}
\tab{5}\spadcommand{ex := discreteLog(y)\bound{ex }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch18}
\begin{paste}{ugxProblemFinitePrimePageFull18}{ugxProblemFinitePrimePageEmpty18}
\pastebutton{ugxProblemFinitePrimePageFull18}{\hidepaste}
\tab{5}\spadcommand{pe ** ex\free{ex pe }}
\indentrel{3}\begin{verbatim}
   (17)  37
                                   Type: PrimeField 101
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty18}
\begin{paste}{ugxProblemFinitePrimePageEmpty18}{ugxProblemFinitePrimePagePatch18}
\pastebutton{ugxProblemFinitePrimePageEmpty18}{\showpaste}
\tab{5}\spadcommand{pe ** ex\free{ex pe }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch19}
\begin{paste}{ugxProblemFinitePrimePageFull19}{ugxProblemFinitePrimePageEmpty19}
\pastebutton{ugxProblemFinitePrimePageFull19}{\hidepaste}
\tab{5}\spadcommand{order y\free{y }}
\indentrel{3}\begin{verbatim}
   (18)  25
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty19}
\begin{paste}{ugxProblemFinitePrimePageEmpty19}{ugxProblemFinitePrimePagePatch19}
\pastebutton{ugxProblemFinitePrimePageEmpty19}{\showpaste}
\tab{5}\spadcommand{order y\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePagePatch20}
\begin{paste}{ugxProblemFinitePrimePageFull20}{ugxProblemFinitePrimePageEmpty20}
\pastebutton{ugxProblemFinitePrimePageFull20}{\hidepaste}
\tab{5}\spadcommand{order pe\free{pe }}
\indentrel{3}\begin{verbatim}
   (19)  100
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFinitePrimePageEmpty20}
\begin{paste}{ugxProblemFinitePrimePageEmpty20}{ugxProblemFinitePrimePagePatch20}
\pastebutton{ugxProblemFinitePrimePageEmpty20}{\showpaste}
\tab{5}\spadcommand{order pe\free{pe }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch1}
\begin{paste}{ugxProblemFiniteCyclicPageFull1}{ugxProblemFiniteCyclicPageEmpty1}
\pastebutton{ugxProblemFiniteCyclicPageFull1}{\hidepaste}
\tab{5}\spadcommand{GF81 := FFCG(3,4);\bound{GF81 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty1}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty1}{ugxProblemFiniteCyclicPagePatch1}
\pastebutton{ugxProblemFiniteCyclicPageEmpty1}{\showpaste}
\tab{5}\spadcommand{GF81 := FFCG(3,4);\bound{GF81 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch2}
\begin{paste}{ugxProblemFiniteCyclicPageFull2}{ugxProblemFiniteCyclicPageEmpty2}
\pastebutton{ugxProblemFiniteCyclicPageFull2}{\hidepaste}
\tab{5}\spadcommand{a := primitiveElement()$GF81\bound{a }\free{GF81 }}
\indentrel{3}\begin{verbatim}
           1
   (2)  %BF
                      Type: FiniteFieldCyclicGroup(3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty2}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty2}{ugxProblemFiniteCyclicPagePatch2}
\pastebutton{ugxProblemFiniteCyclicPageEmpty2}{\showpaste}
\tab{5}\spadcommand{a := primitiveElement()$GF81\bound{a }\free{GF81 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch3}
\begin{paste}{ugxProblemFiniteCyclicPageFull3}{ugxProblemFiniteCyclicPageEmpty3}
\pastebutton{ugxProblemFiniteCyclicPageFull3}{\hidepaste}
\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
\indentrel{3}\begin{verbatim}
           72
   (3)  %BF
                      Type: FiniteFieldCyclicGroup(3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty3}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty3}{ugxProblemFiniteCyclicPagePatch3}
\pastebutton{ugxProblemFiniteCyclicPageEmpty3}{\showpaste}
\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch4}
\begin{paste}{ugxProblemFiniteCyclicPageFull4}{ugxProblemFiniteCyclicPageEmpty4}
\pastebutton{ugxProblemFiniteCyclicPageFull4}{\hidepaste}
\tab{5}\spadcommand{b\free{b }}
\indentrel{3}\begin{verbatim}
           72
   (4)  %BF
                      Type: FiniteFieldCyclicGroup(3,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty4}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty4}{ugxProblemFiniteCyclicPagePatch4}
\pastebutton{ugxProblemFiniteCyclicPageEmpty4}{\showpaste}
\tab{5}\spadcommand{b\free{b }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch5}
\begin{paste}{ugxProblemFiniteCyclicPageFull5}{ugxProblemFiniteCyclicPageEmpty5}
\pastebutton{ugxProblemFiniteCyclicPageFull5}{\hidepaste}
\tab{5}\spadcommand{discreteLog b\free{b }}
\indentrel{3}\begin{verbatim}
   (5)  72
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty5}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty5}{ugxProblemFiniteCyclicPagePatch5}
\pastebutton{ugxProblemFiniteCyclicPageEmpty5}{\showpaste}
\tab{5}\spadcommand{discreteLog b\free{b }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch6}
\begin{paste}{ugxProblemFiniteCyclicPageFull6}{ugxProblemFiniteCyclicPageEmpty6}
\pastebutton{ugxProblemFiniteCyclicPageFull6}{\hidepaste}
\tab{5}\spadcommand{GF9 := FF(3,2);\bound{GF9 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty6}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty6}{ugxProblemFiniteCyclicPagePatch6}
\pastebutton{ugxProblemFiniteCyclicPageEmpty6}{\showpaste}
\tab{5}\spadcommand{GF9 := FF(3,2);\bound{GF9 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch7}
\begin{paste}{ugxProblemFiniteCyclicPageFull7}{ugxProblemFiniteCyclicPageEmpty7}
\pastebutton{ugxProblemFiniteCyclicPageFull7}{\hidepaste}
\tab{5}\spadcommand{GF729 := FFCGX(GF9,3);\bound{GF729 }\free{GF9 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty7}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty7}{ugxProblemFiniteCyclicPagePatch7}
\pastebutton{ugxProblemFiniteCyclicPageEmpty7}{\showpaste}
\tab{5}\spadcommand{GF729 := FFCGX(GF9,3);\bound{GF729 }\free{GF9 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch8}
\begin{paste}{ugxProblemFiniteCyclicPageFull8}{ugxProblemFiniteCyclicPageEmpty8}
\pastebutton{ugxProblemFiniteCyclicPageFull8}{\hidepaste}
\tab{5}\spadcommand{r := (random()$GF729) ** 20\free{GF729 }\bound{r }}
\indentrel{3}\begin{verbatim}
           68
   (8)  %CN
Type: FiniteFieldCyclicGroupExtension(FiniteField(3,2),3)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty8}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty8}{ugxProblemFiniteCyclicPagePatch8}
\pastebutton{ugxProblemFiniteCyclicPageEmpty8}{\showpaste}
\tab{5}\spadcommand{r := (random()$GF729) ** 20\free{GF729 }\bound{r }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch9}
\begin{paste}{ugxProblemFiniteCyclicPageFull9}{ugxProblemFiniteCyclicPageEmpty9}
\pastebutton{ugxProblemFiniteCyclicPageFull9}{\hidepaste}
\tab{5}\spadcommand{trace(r)\free{r }}
\indentrel{3}\begin{verbatim}
   (9)  2
                                 Type: FiniteField(3,2)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty9}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty9}{ugxProblemFiniteCyclicPagePatch9}
\pastebutton{ugxProblemFiniteCyclicPageEmpty9}{\showpaste}
\tab{5}\spadcommand{trace(r)\free{r }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch10}
\begin{paste}{ugxProblemFiniteCyclicPageFull10}{ugxProblemFiniteCyclicPageEmpty10}
\pastebutton{ugxProblemFiniteCyclicPageFull10}{\hidepaste}
\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty10}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty10}{ugxProblemFiniteCyclicPagePatch10}
\pastebutton{ugxProblemFiniteCyclicPageEmpty10}{\showpaste}
\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch11}
\begin{paste}{ugxProblemFiniteCyclicPageFull11}{ugxProblemFiniteCyclicPageEmpty11}
\pastebutton{ugxProblemFiniteCyclicPageFull11}{\hidepaste}
\tab{5}\spadcommand{f := createPrimitivePoly(4)$FFPOLY(GF3)\bound{f }\free{GF3 }}
\indentrel{3}\begin{verbatim}
          4
   (11)  ?  + ? + 2
          Type: SparseUnivariatePolynomial PrimeField 3
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty11}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty11}{ugxProblemFiniteCyclicPagePatch11}
\pastebutton{ugxProblemFiniteCyclicPageEmpty11}{\showpaste}
\tab{5}\spadcommand{f := createPrimitivePoly(4)$FFPOLY(GF3)\bound{f }\free{GF3 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch12}
\begin{paste}{ugxProblemFiniteCyclicPageFull12}{ugxProblemFiniteCyclicPageEmpty12}
\pastebutton{ugxProblemFiniteCyclicPageFull12}{\hidepaste}
\tab{5}\spadcommand{GF81 := FFCGP(GF3,f);\bound{GF81x }\free{f GF3 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty12}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty12}{ugxProblemFiniteCyclicPagePatch12}
\pastebutton{ugxProblemFiniteCyclicPageEmpty12}{\showpaste}
\tab{5}\spadcommand{GF81 := FFCGP(GF3,f);\bound{GF81x }\free{f GF3 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPagePatch13}
\begin{paste}{ugxProblemFiniteCyclicPageFull13}{ugxProblemFiniteCyclicPageEmpty13}
\pastebutton{ugxProblemFiniteCyclicPageFull13}{\hidepaste}
\tab{5}\spadcommand{random()$GF81\free{GF81x }}
\indentrel{3}\begin{verbatim}
            8
   (13)  %BF
Type: FiniteFieldCyclicGroupExtensionByPolynomial(PrimeField 3,?**4+?+2)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteCyclicPageEmpty13}
\begin{paste}{ugxProblemFiniteCyclicPageEmpty13}{ugxProblemFiniteCyclicPagePatch13}
\pastebutton{ugxProblemFiniteCyclicPageEmpty13}{\showpaste}
\tab{5}\spadcommand{random()$GF81\free{GF81x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePagePatch1}
\begin{paste}{ugxProblemSeriesCreatePageFull1}{ugxProblemSeriesCreatePageEmpty1}
\pastebutton{ugxProblemSeriesCreatePageFull1}{\hidepaste}
\tab{5}\spadcommand{x := series 'x\bound{x }}
\indentrel{3}\begin{verbatim}
   (1)  x
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePageEmpty1}
\begin{paste}{ugxProblemSeriesCreatePageEmpty1}{ugxProblemSeriesCreatePagePatch1}
\pastebutton{ugxProblemSeriesCreatePageEmpty1}{\showpaste}
\tab{5}\spadcommand{x := series 'x\bound{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePagePatch2}
\begin{paste}{ugxProblemSeriesCreatePageFull2}{ugxProblemSeriesCreatePageEmpty2}
\pastebutton{ugxProblemSeriesCreatePageFull2}{\hidepaste}
\tab{5}\spadcommand{1/(1 - x - x**2)\free{x }}
\indentrel{3}\begin{verbatim}
   (2)
               2     3     4     5      6      7      8
     1 + x + 2x  + 3x  + 5x  + 8x  + 13x  + 21x  + 34x
   + 
        9      10      11
     55x  + 89x   + O(x  )
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePageEmpty2}
\begin{paste}{ugxProblemSeriesCreatePageEmpty2}{ugxProblemSeriesCreatePagePatch2}
\pastebutton{ugxProblemSeriesCreatePageEmpty2}{\showpaste}
\tab{5}\spadcommand{1/(1 - x - x**2)\free{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePagePatch3}
\begin{paste}{ugxProblemSeriesCreatePageFull3}{ugxProblemSeriesCreatePageEmpty3}
\pastebutton{ugxProblemSeriesCreatePageFull3}{\hidepaste}
\tab{5}\spadcommand{sin(x)\free{x }}
\indentrel{3}\begin{verbatim}
   (3)
         1  3    1   5     1   7      1    9
     x - Ä x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
         6      120      5040      362880
   + 
           1     11      12
     - ÄÄÄÄÄÄÄÄ x   + O(x  )
       39916800
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePageEmpty3}
\begin{paste}{ugxProblemSeriesCreatePageEmpty3}{ugxProblemSeriesCreatePagePatch3}
\pastebutton{ugxProblemSeriesCreatePageEmpty3}{\showpaste}
\tab{5}\spadcommand{sin(x)\free{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePagePatch4}
\begin{paste}{ugxProblemSeriesCreatePageFull4}{ugxProblemSeriesCreatePageEmpty4}
\pastebutton{ugxProblemSeriesCreatePageFull4}{\hidepaste}
\tab{5}\spadcommand{sin(1 + x)\free{x }}
\indentrel{3}\begin{verbatim}
   (4)
                        sin(1)  2   cos(1)  3   sin(1)  4
     sin(1) + cos(1)x - ÄÄÄÄÄÄ x  - ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄ x
                           2           6          24
   + 
     cos(1)  5   sin(1)  6   cos(1)  7   sin(1)  8
     ÄÄÄÄÄÄ x  - ÄÄÄÄÄÄ x  - ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄ x
       120         720        5040        40320
   + 
     cos(1)  9    sin(1)  10      11
     ÄÄÄÄÄÄ x  - ÄÄÄÄÄÄÄ x   + O(x  )
     362880      3628800
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePageEmpty4}
\begin{paste}{ugxProblemSeriesCreatePageEmpty4}{ugxProblemSeriesCreatePagePatch4}
\pastebutton{ugxProblemSeriesCreatePageEmpty4}{\showpaste}
\tab{5}\spadcommand{sin(1 + x)\free{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePagePatch5}
\begin{paste}{ugxProblemSeriesCreatePageFull5}{ugxProblemSeriesCreatePageEmpty5}
\pastebutton{ugxProblemSeriesCreatePageFull5}{\hidepaste}
\tab{5}\spadcommand{sin(a * x)\free{x }}
\indentrel{3}\begin{verbatim}
   (5)
            3        5        7          9
           a   3    a   5    a    7     a     9
     a x - ÄÄ x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
            6      120      5040      362880
   + 
           11
          a      11      12
     - ÄÄÄÄÄÄÄÄ x   + O(x  )
       39916800
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePageEmpty5}
\begin{paste}{ugxProblemSeriesCreatePageEmpty5}{ugxProblemSeriesCreatePagePatch5}
\pastebutton{ugxProblemSeriesCreatePageEmpty5}{\showpaste}
\tab{5}\spadcommand{sin(a * x)\free{x }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePagePatch6}
\begin{paste}{ugxProblemSeriesCreatePageFull6}{ugxProblemSeriesCreatePageEmpty6}
\pastebutton{ugxProblemSeriesCreatePageFull6}{\hidepaste}
\tab{5}\spadcommand{series(1/log(y),y = 1)}
\indentrel{3}\begin{verbatim}
   (6)
            - 1   1    1            1        2
     (y - 1)    + Ä - ÄÄ (y - 1) + ÄÄ (y - 1)
                  2   12           24
   + 
        19        3    3         4    863         5
     - ÄÄÄ (y - 1)  + ÄÄÄ (y - 1)  - ÄÄÄÄÄ (y - 1)
       720            160            60480
   + 
      275         6    33953         7     8183         8
     ÄÄÄÄÄ (y - 1)  - ÄÄÄÄÄÄÄ (y - 1)  + ÄÄÄÄÄÄÄ (y - 1)
     24192            3628800            1036800
   + 
        3250433         9            10
     - ÄÄÄÄÄÄÄÄÄ (y - 1)  + O((y - 1)  )
       479001600
  Type: UnivariatePuiseuxSeries(Expression Integer,y,1)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePageEmpty6}
\begin{paste}{ugxProblemSeriesCreatePageEmpty6}{ugxProblemSeriesCreatePagePatch6}
\pastebutton{ugxProblemSeriesCreatePageEmpty6}{\showpaste}
\tab{5}\spadcommand{series(1/log(y),y = 1)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePagePatch7}
\begin{paste}{ugxProblemSeriesCreatePageFull7}{ugxProblemSeriesCreatePageEmpty7}
\pastebutton{ugxProblemSeriesCreatePageFull7}{\hidepaste}
\tab{5}\spadcommand{y : UTS(FLOAT,'z,0) := exp(z)\bound{y }}
\indentrel{3}\begin{verbatim}
   (7)
                    2                            3
     1.0 + z + 0.5 z  + 0.1666666666 6666666667 z
   + 
                                4
     0.0416666666 6666666666 7 z
   + 
                                 5
     0.0083333333 3333333333 34 z
   + 
                                 6
     0.0013888888 8888888888 89 z
   + 
                                 7
     0.0001984126 9841269841 27 z
   + 
                                   8
     0.0000248015 8730158730 1587 z
   + 
                                    9
     0.0000027557 3192239858 90653 z
   + 
                                   10      11
     0.2755731922 3985890653 E -6 z   + O(z  )
              Type: UnivariateTaylorSeries(Float,z,0.0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePageEmpty7}
\begin{paste}{ugxProblemSeriesCreatePageEmpty7}{ugxProblemSeriesCreatePagePatch7}
\pastebutton{ugxProblemSeriesCreatePageEmpty7}{\showpaste}
\tab{5}\spadcommand{y : UTS(FLOAT,'z,0) := exp(z)\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePagePatch8}
\begin{paste}{ugxProblemSeriesCreatePageFull8}{ugxProblemSeriesCreatePageEmpty8}
\pastebutton{ugxProblemSeriesCreatePageFull8}{\hidepaste}
\tab{5}\spadcommand{series(1/factorial(n),n,w = 0)}
\indentrel{3}\begin{verbatim}
   (8)
             1  2   1  3    1  4    1   5    1   6
     1 + w + Ä w  + Ä w  + ÄÄ w  + ÄÄÄ w  + ÄÄÄ w
             2      6      24      120      720
   + 
     1   7     1    8      1    9      1     10      11
   ÄÄÄÄ w  + ÄÄÄÄÄ w  + ÄÄÄÄÄÄ w  + ÄÄÄÄÄÄÄ w   + O(w  )
   5040      40320      362880      3628800
  Type: UnivariatePuiseuxSeries(Expression Integer,w,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesCreatePageEmpty8}
\begin{paste}{ugxProblemSeriesCreatePageEmpty8}{ugxProblemSeriesCreatePagePatch8}
\pastebutton{ugxProblemSeriesCreatePageEmpty8}{\showpaste}
\tab{5}\spadcommand{series(1/factorial(n),n,w = 0)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch1}
\begin{paste}{ugxProblemFiniteNormalPageFull1}{ugxProblemFiniteNormalPageEmpty1}
\pastebutton{ugxProblemFiniteNormalPageFull1}{\hidepaste}
\tab{5}\spadcommand{K := FFNB(3,8)\bound{K }}
\indentrel{3}\begin{verbatim}
   (1)  FiniteFieldNormalBasis(3,8)
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty1}
\begin{paste}{ugxProblemFiniteNormalPageEmpty1}{ugxProblemFiniteNormalPagePatch1}
\pastebutton{ugxProblemFiniteNormalPageEmpty1}{\showpaste}
\tab{5}\spadcommand{K := FFNB(3,8)\bound{K }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch2}
\begin{paste}{ugxProblemFiniteNormalPageFull2}{ugxProblemFiniteNormalPageEmpty2}
\pastebutton{ugxProblemFiniteNormalPageFull2}{\hidepaste}
\tab{5}\spadcommand{a := normalElement()$K\bound{a }\free{K }}
\indentrel{3}\begin{verbatim}
   (2)  %CO
                      Type: FiniteFieldNormalBasis(3,8)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty2}
\begin{paste}{ugxProblemFiniteNormalPageEmpty2}{ugxProblemFiniteNormalPagePatch2}
\pastebutton{ugxProblemFiniteNormalPageEmpty2}{\showpaste}
\tab{5}\spadcommand{a := normalElement()$K\bound{a }\free{K }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch3}
\begin{paste}{ugxProblemFiniteNormalPageFull3}{ugxProblemFiniteNormalPageEmpty3}
\pastebutton{ugxProblemFiniteNormalPageFull3}{\hidepaste}
\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
\indentrel{3}\begin{verbatim}
             7       5
            q       q       q
   (3)  2%CO   + %CO   + %CO
                      Type: FiniteFieldNormalBasis(3,8)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty3}
\begin{paste}{ugxProblemFiniteNormalPageEmpty3}{ugxProblemFiniteNormalPagePatch3}
\pastebutton{ugxProblemFiniteNormalPageEmpty3}{\showpaste}
\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch4}
\begin{paste}{ugxProblemFiniteNormalPageFull4}{ugxProblemFiniteNormalPageEmpty4}
\pastebutton{ugxProblemFiniteNormalPageFull4}{\hidepaste}
\tab{5}\spadcommand{GF9 := FFNB(3,2);\bound{GF9 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty4}
\begin{paste}{ugxProblemFiniteNormalPageEmpty4}{ugxProblemFiniteNormalPagePatch4}
\pastebutton{ugxProblemFiniteNormalPageEmpty4}{\showpaste}
\tab{5}\spadcommand{GF9 := FFNB(3,2);\bound{GF9 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch5}
\begin{paste}{ugxProblemFiniteNormalPageFull5}{ugxProblemFiniteNormalPageEmpty5}
\pastebutton{ugxProblemFiniteNormalPageFull5}{\hidepaste}
\tab{5}\spadcommand{GF729 := FFNBX(GF9,3);\bound{GF729 }\free{GF9 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty5}
\begin{paste}{ugxProblemFiniteNormalPageEmpty5}{ugxProblemFiniteNormalPagePatch5}
\pastebutton{ugxProblemFiniteNormalPageEmpty5}{\showpaste}
\tab{5}\spadcommand{GF729 := FFNBX(GF9,3);\bound{GF729 }\free{GF9 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch6}
\begin{paste}{ugxProblemFiniteNormalPageFull6}{ugxProblemFiniteNormalPageEmpty6}
\pastebutton{ugxProblemFiniteNormalPageFull6}{\hidepaste}
\tab{5}\spadcommand{r := random()$GF729\bound{r }\free{GF729 }}
\indentrel{3}\begin{verbatim}
                 2
            q   q
   (6)  2%CP %CQ
Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty6}
\begin{paste}{ugxProblemFiniteNormalPageEmpty6}{ugxProblemFiniteNormalPagePatch6}
\pastebutton{ugxProblemFiniteNormalPageEmpty6}{\showpaste}
\tab{5}\spadcommand{r := random()$GF729\bound{r }\free{GF729 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch7}
\begin{paste}{ugxProblemFiniteNormalPageFull7}{ugxProblemFiniteNormalPageEmpty7}
\pastebutton{ugxProblemFiniteNormalPageFull7}{\hidepaste}
\tab{5}\spadcommand{r + r**3 + r**9 + r**27\free{r }}
\indentrel{3}\begin{verbatim}
                          2
             q           q            q       q
   (7)  (2%CP  + 2%CP)%CQ   + 2%CP %CQ  + 2%CP %CQ
Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty7}
\begin{paste}{ugxProblemFiniteNormalPageEmpty7}{ugxProblemFiniteNormalPagePatch7}
\pastebutton{ugxProblemFiniteNormalPageEmpty7}{\showpaste}
\tab{5}\spadcommand{r + r**3 + r**9 + r**27\free{r }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch8}
\begin{paste}{ugxProblemFiniteNormalPageFull8}{ugxProblemFiniteNormalPageEmpty8}
\pastebutton{ugxProblemFiniteNormalPageFull8}{\hidepaste}
\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty8}
\begin{paste}{ugxProblemFiniteNormalPageEmpty8}{ugxProblemFiniteNormalPagePatch8}
\pastebutton{ugxProblemFiniteNormalPageEmpty8}{\showpaste}
\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch9}
\begin{paste}{ugxProblemFiniteNormalPageFull9}{ugxProblemFiniteNormalPageEmpty9}
\pastebutton{ugxProblemFiniteNormalPageFull9}{\hidepaste}
\tab{5}\spadcommand{f := createNormalPoly(4)$FFPOLY(GF3)\free{GF3 }\bound{f }}
\indentrel{3}\begin{verbatim}
         4     3
   (9)  ?  + 2?  + 2
          Type: SparseUnivariatePolynomial PrimeField 3
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty9}
\begin{paste}{ugxProblemFiniteNormalPageEmpty9}{ugxProblemFiniteNormalPagePatch9}
\pastebutton{ugxProblemFiniteNormalPageEmpty9}{\showpaste}
\tab{5}\spadcommand{f := createNormalPoly(4)$FFPOLY(GF3)\free{GF3 }\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch10}
\begin{paste}{ugxProblemFiniteNormalPageFull10}{ugxProblemFiniteNormalPageEmpty10}
\pastebutton{ugxProblemFiniteNormalPageFull10}{\hidepaste}
\tab{5}\spadcommand{GF81 := FFNBP(GF3,f);\bound{GF81 }\free{f GF3 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty10}
\begin{paste}{ugxProblemFiniteNormalPageEmpty10}{ugxProblemFiniteNormalPagePatch10}
\pastebutton{ugxProblemFiniteNormalPageEmpty10}{\showpaste}
\tab{5}\spadcommand{GF81 := FFNBP(GF3,f);\bound{GF81 }\free{f GF3 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch11}
\begin{paste}{ugxProblemFiniteNormalPageFull11}{ugxProblemFiniteNormalPageEmpty11}
\pastebutton{ugxProblemFiniteNormalPageFull11}{\hidepaste}
\tab{5}\spadcommand{r := random()$GF81\free{GF81 }\bound{r1 }}
\indentrel{3}\begin{verbatim}
             3       2
            q       q       q
   (11)  %CR   + %CR   + %CR
Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty11}
\begin{paste}{ugxProblemFiniteNormalPageEmpty11}{ugxProblemFiniteNormalPagePatch11}
\pastebutton{ugxProblemFiniteNormalPageEmpty11}{\showpaste}
\tab{5}\spadcommand{r := random()$GF81\free{GF81 }\bound{r1 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch12}
\begin{paste}{ugxProblemFiniteNormalPageFull12}{ugxProblemFiniteNormalPageEmpty12}
\pastebutton{ugxProblemFiniteNormalPageFull12}{\hidepaste}
\tab{5}\spadcommand{r * r**3 * r**9 * r**27\free{r1 }}
\indentrel{3}\begin{verbatim}
              3        2
             q        q        q
   (12)  2%CR   + 2%CR   + 2%CR  + 2%CR
Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty12}
\begin{paste}{ugxProblemFiniteNormalPageEmpty12}{ugxProblemFiniteNormalPagePatch12}
\pastebutton{ugxProblemFiniteNormalPageEmpty12}{\showpaste}
\tab{5}\spadcommand{r * r**3 * r**9 * r**27\free{r1 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPagePatch13}
\begin{paste}{ugxProblemFiniteNormalPageFull13}{ugxProblemFiniteNormalPageEmpty13}
\pastebutton{ugxProblemFiniteNormalPageFull13}{\hidepaste}
\tab{5}\spadcommand{norm r\free{r1 }}
\indentrel{3}\begin{verbatim}
   (13)  2
                                     Type: PrimeField 3
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteNormalPageEmpty13}
\begin{paste}{ugxProblemFiniteNormalPageEmpty13}{ugxProblemFiniteNormalPagePatch13}
\pastebutton{ugxProblemFiniteNormalPageEmpty13}{\showpaste}
\tab{5}\spadcommand{norm r\free{r1 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPagePatch1}
\begin{paste}{ugxProblemSeriesFormulaPageFull1}{ugxProblemSeriesFormulaPageEmpty1}
\pastebutton{ugxProblemSeriesFormulaPageFull1}{\hidepaste}
\tab{5}\spadcommand{series(n +-> 1/factorial(n),x = 0)}
\indentrel{3}\begin{verbatim}
   (1)
             1  2   1  3    1  4    1   5    1   6
     1 + x + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
             2      6      24      120      720
   + 
     1   7     1    8      1    9      1     10      11
   ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x   + O(x  )
   5040      40320      362880      3628800
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPageEmpty1}
\begin{paste}{ugxProblemSeriesFormulaPageEmpty1}{ugxProblemSeriesFormulaPagePatch1}
\pastebutton{ugxProblemSeriesFormulaPageEmpty1}{\showpaste}
\tab{5}\spadcommand{series(n +-> 1/factorial(n),x = 0)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPagePatch2}
\begin{paste}{ugxProblemSeriesFormulaPageFull2}{ugxProblemSeriesFormulaPageEmpty2}
\pastebutton{ugxProblemSeriesFormulaPageFull2}{\hidepaste}
\tab{5}\spadcommand{series(n +-> (-1)**(n-1)/n,x = 1,1..)}
\indentrel{3}\begin{verbatim}
   (2)
               1        2   1        3   1        4
     (x - 1) - Ä (x - 1)  + Ä (x - 1)  - Ä (x - 1)
               2            3            4
   + 
     1        5   1        6   1        7   1        8
     Ä (x - 1)  - Ä (x - 1)  + Ä (x - 1)  - Ä (x - 1)
     5            6            7            8
   + 
   1        9    1        10    1        11            12
   Ä (x - 1)  - ÄÄ (x - 1)   + ÄÄ (x - 1)   + O((x - 1)  )
   9            10             11
  Type: UnivariatePuiseuxSeries(Expression Integer,x,1)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPageEmpty2}
\begin{paste}{ugxProblemSeriesFormulaPageEmpty2}{ugxProblemSeriesFormulaPagePatch2}
\pastebutton{ugxProblemSeriesFormulaPageEmpty2}{\showpaste}
\tab{5}\spadcommand{series(n +-> (-1)**(n-1)/n,x = 1,1..)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPagePatch3}
\begin{paste}{ugxProblemSeriesFormulaPageFull3}{ugxProblemSeriesFormulaPageEmpty3}
\pastebutton{ugxProblemSeriesFormulaPageFull3}{\hidepaste}
\tab{5}\spadcommand{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)}
\indentrel{3}\begin{verbatim}
   (3)
         1  3    1   5     1   7      1    9
     x - Ä x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
         6      120      5040      362880
   + 
           1     11      12
     - ÄÄÄÄÄÄÄÄ x   + O(x  )
       39916800
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPageEmpty3}
\begin{paste}{ugxProblemSeriesFormulaPageEmpty3}{ugxProblemSeriesFormulaPagePatch3}
\pastebutton{ugxProblemSeriesFormulaPageEmpty3}{\showpaste}
\tab{5}\spadcommand{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPagePatch4}
\begin{paste}{ugxProblemSeriesFormulaPageFull4}{ugxProblemSeriesFormulaPageEmpty4}
\pastebutton{ugxProblemSeriesFormulaPageFull4}{\hidepaste}
\tab{5}\spadcommand{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)}
\indentrel{3}\begin{verbatim}
   (4)
      1              5         7
      Ä              Ä         Ä
      3   1      1   3     1   3      1    3
     x  - Ä x + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
          6     120      5040      362880
   + 
                 11
                 ÄÄ
           1      3      4
     - ÄÄÄÄÄÄÄÄ x   + O(x )
       39916800
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPageEmpty4}
\begin{paste}{ugxProblemSeriesFormulaPageEmpty4}{ugxProblemSeriesFormulaPagePatch4}
\pastebutton{ugxProblemSeriesFormulaPageEmpty4}{\showpaste}
\tab{5}\spadcommand{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPagePatch5}
\begin{paste}{ugxProblemSeriesFormulaPageFull5}{ugxProblemSeriesFormulaPageEmpty5}
\pastebutton{ugxProblemSeriesFormulaPageFull5}{\hidepaste}
\tab{5}\spadcommand{cscx := series(n +-> (-1)**((n-1)/2) * 2 * (2**n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2)\bound{cscx }}
\indentrel{3}\begin{verbatim}
   (5)
      - 1   1      7   3     31   5     127   7
     x    + Ä x + ÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x
            6     360      15120      604800
   + 
        73    9      10
     ÄÄÄÄÄÄÄ x  + O(x  )
     3421440
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPageEmpty5}
\begin{paste}{ugxProblemSeriesFormulaPageEmpty5}{ugxProblemSeriesFormulaPagePatch5}
\pastebutton{ugxProblemSeriesFormulaPageEmpty5}{\showpaste}
\tab{5}\spadcommand{cscx := series(n +-> (-1)**((n-1)/2) * 2 * (2**n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2)\bound{cscx }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPagePatch6}
\begin{paste}{ugxProblemSeriesFormulaPageFull6}{ugxProblemSeriesFormulaPageEmpty6}
\pastebutton{ugxProblemSeriesFormulaPageFull6}{\hidepaste}
\tab{5}\spadcommand{1/cscx\free{cscx }}
\indentrel{3}\begin{verbatim}
   (6)
         1  3    1   5     1   7      1    9
     x - Ä x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
         6      120      5040      362880
   + 
           1     11      12
     - ÄÄÄÄÄÄÄÄ x   + O(x  )
       39916800
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPageEmpty6}
\begin{paste}{ugxProblemSeriesFormulaPageEmpty6}{ugxProblemSeriesFormulaPagePatch6}
\pastebutton{ugxProblemSeriesFormulaPageEmpty6}{\showpaste}
\tab{5}\spadcommand{1/cscx\free{cscx }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPagePatch7}
\begin{paste}{ugxProblemSeriesFormulaPageFull7}{ugxProblemSeriesFormulaPageEmpty7}
\pastebutton{ugxProblemSeriesFormulaPageFull7}{\hidepaste}
\tab{5}\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2)\bound{asinx }}
\indentrel{3}\begin{verbatim}
   (7)
       1  3    3  5    5   7    35   9    63   11      12
   x + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄÄ x  + ÄÄÄÄ x   + O(x  )
       6      40      112      1152      2816
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPageEmpty7}
\begin{paste}{ugxProblemSeriesFormulaPageEmpty7}{ugxProblemSeriesFormulaPagePatch7}
\pastebutton{ugxProblemSeriesFormulaPageEmpty7}{\showpaste}
\tab{5}\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2)\bound{asinx }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPagePatch8}
\begin{paste}{ugxProblemSeriesFormulaPageFull8}{ugxProblemSeriesFormulaPageEmpty8}
\pastebutton{ugxProblemSeriesFormulaPageFull8}{\hidepaste}
\tab{5}\spadcommand{sin(asinx)\free{asinx }}
\indentrel{3}\begin{verbatim}
               12
   (8)  x + O(x  )
  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemSeriesFormulaPageEmpty8}
\begin{paste}{ugxProblemSeriesFormulaPageEmpty8}{ugxProblemSeriesFormulaPagePatch8}
\pastebutton{ugxProblemSeriesFormulaPageEmpty8}{\showpaste}
\tab{5}\spadcommand{sin(asinx)\free{asinx }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch1}
\begin{paste}{ugxProblemFiniteModulusPageFull1}{ugxProblemFiniteModulusPageEmpty1}
\pastebutton{ugxProblemFiniteModulusPageFull1}{\hidepaste}
\tab{5}\spadcommand{GF4096 := FF(2,12);\bound{GF4096 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty1}
\begin{paste}{ugxProblemFiniteModulusPageEmpty1}{ugxProblemFiniteModulusPagePatch1}
\pastebutton{ugxProblemFiniteModulusPageEmpty1}{\showpaste}
\tab{5}\spadcommand{GF4096 := FF(2,12);\bound{GF4096 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch2}
\begin{paste}{ugxProblemFiniteModulusPageFull2}{ugxProblemFiniteModulusPageEmpty2}
\pastebutton{ugxProblemFiniteModulusPageFull2}{\hidepaste}
\tab{5}\spadcommand{a := index(2)$GF4096\bound{a }\free{GF4096 }}
\indentrel{3}\begin{verbatim}
   (2)  %CS
                                Type: FiniteField(2,12)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty2}
\begin{paste}{ugxProblemFiniteModulusPageEmpty2}{ugxProblemFiniteModulusPagePatch2}
\pastebutton{ugxProblemFiniteModulusPageEmpty2}{\showpaste}
\tab{5}\spadcommand{a := index(2)$GF4096\bound{a }\free{GF4096 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch3}
\begin{paste}{ugxProblemFiniteModulusPageFull3}{ugxProblemFiniteModulusPageEmpty3}
\pastebutton{ugxProblemFiniteModulusPageFull3}{\hidepaste}
\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
\indentrel{3}\begin{verbatim}
           5      3
   (3)  %CS  + %CS  + %CS + 1
                                Type: FiniteField(2,12)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty3}
\begin{paste}{ugxProblemFiniteModulusPageEmpty3}{ugxProblemFiniteModulusPagePatch3}
\pastebutton{ugxProblemFiniteModulusPageEmpty3}{\showpaste}
\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch4}
\begin{paste}{ugxProblemFiniteModulusPageFull4}{ugxProblemFiniteModulusPageEmpty4}
\pastebutton{ugxProblemFiniteModulusPageFull4}{\hidepaste}
\tab{5}\spadcommand{b ** 1000\free{b }}
\indentrel{3}\begin{verbatim}
           10      9      7      5      4      3
   (4)  %CS   + %CS  + %CS  + %CS  + %CS  + %CS  + %CS
                                Type: FiniteField(2,12)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty4}
\begin{paste}{ugxProblemFiniteModulusPageEmpty4}{ugxProblemFiniteModulusPagePatch4}
\pastebutton{ugxProblemFiniteModulusPageEmpty4}{\showpaste}
\tab{5}\spadcommand{b ** 1000\free{b }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch5}
\begin{paste}{ugxProblemFiniteModulusPageFull5}{ugxProblemFiniteModulusPageEmpty5}
\pastebutton{ugxProblemFiniteModulusPageFull5}{\hidepaste}
\tab{5}\spadcommand{c := a/b\free{a b }\bound{c }}
\indentrel{3}\begin{verbatim}
           11      8      7      5      4      3      2
   (5)  %CS   + %CS  + %CS  + %CS  + %CS  + %CS  + %CS
                                Type: FiniteField(2,12)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty5}
\begin{paste}{ugxProblemFiniteModulusPageEmpty5}{ugxProblemFiniteModulusPagePatch5}
\pastebutton{ugxProblemFiniteModulusPageEmpty5}{\showpaste}
\tab{5}\spadcommand{c := a/b\free{a b }\bound{c }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch6}
\begin{paste}{ugxProblemFiniteModulusPageFull6}{ugxProblemFiniteModulusPageEmpty6}
\pastebutton{ugxProblemFiniteModulusPageFull6}{\hidepaste}
\tab{5}\spadcommand{norm c\free{c }}
\indentrel{3}\begin{verbatim}
   (6)  1
                                     Type: PrimeField 2
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty6}
\begin{paste}{ugxProblemFiniteModulusPageEmpty6}{ugxProblemFiniteModulusPagePatch6}
\pastebutton{ugxProblemFiniteModulusPageEmpty6}{\showpaste}
\tab{5}\spadcommand{norm c\free{c }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch7}
\begin{paste}{ugxProblemFiniteModulusPageFull7}{ugxProblemFiniteModulusPageEmpty7}
\pastebutton{ugxProblemFiniteModulusPageFull7}{\hidepaste}
\tab{5}\spadcommand{trace c\free{c }}
\indentrel{3}\begin{verbatim}
   (7)  0
                                     Type: PrimeField 2
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty7}
\begin{paste}{ugxProblemFiniteModulusPageEmpty7}{ugxProblemFiniteModulusPagePatch7}
\pastebutton{ugxProblemFiniteModulusPageEmpty7}{\showpaste}
\tab{5}\spadcommand{trace c\free{c }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch8}
\begin{paste}{ugxProblemFiniteModulusPageFull8}{ugxProblemFiniteModulusPageEmpty8}
\pastebutton{ugxProblemFiniteModulusPageFull8}{\hidepaste}
\tab{5}\spadcommand{dL := discreteLog a\free{a }\bound{dL }}
\indentrel{3}\begin{verbatim}
   (8)  1729
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty8}
\begin{paste}{ugxProblemFiniteModulusPageEmpty8}{ugxProblemFiniteModulusPagePatch8}
\pastebutton{ugxProblemFiniteModulusPageEmpty8}{\showpaste}
\tab{5}\spadcommand{dL := discreteLog a\free{a }\bound{dL }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch9}
\begin{paste}{ugxProblemFiniteModulusPageFull9}{ugxProblemFiniteModulusPageEmpty9}
\pastebutton{ugxProblemFiniteModulusPageFull9}{\hidepaste}
\tab{5}\spadcommand{g ** dL\free{dL g }}
\indentrel{3}\begin{verbatim}
         1729
   (9)  g
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty9}
\begin{paste}{ugxProblemFiniteModulusPageEmpty9}{ugxProblemFiniteModulusPagePatch9}
\pastebutton{ugxProblemFiniteModulusPageEmpty9}{\showpaste}
\tab{5}\spadcommand{g ** dL\free{dL g }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch10}
\begin{paste}{ugxProblemFiniteModulusPageFull10}{ugxProblemFiniteModulusPageEmpty10}
\pastebutton{ugxProblemFiniteModulusPageFull10}{\hidepaste}
\tab{5}\spadcommand{GF16 := FF(2,4);\bound{GF16 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty10}
\begin{paste}{ugxProblemFiniteModulusPageEmpty10}{ugxProblemFiniteModulusPagePatch10}
\pastebutton{ugxProblemFiniteModulusPageEmpty10}{\showpaste}
\tab{5}\spadcommand{GF16 := FF(2,4);\bound{GF16 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch11}
\begin{paste}{ugxProblemFiniteModulusPageFull11}{ugxProblemFiniteModulusPageEmpty11}
\pastebutton{ugxProblemFiniteModulusPageFull11}{\hidepaste}
\tab{5}\spadcommand{GF4096 := FFX(GF16,3);\bound{GF4096x }\free{GF16 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty11}
\begin{paste}{ugxProblemFiniteModulusPageEmpty11}{ugxProblemFiniteModulusPagePatch11}
\pastebutton{ugxProblemFiniteModulusPageEmpty11}{\showpaste}
\tab{5}\spadcommand{GF4096 := FFX(GF16,3);\bound{GF4096x }\free{GF16 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch12}
\begin{paste}{ugxProblemFiniteModulusPageFull12}{ugxProblemFiniteModulusPageEmpty12}
\pastebutton{ugxProblemFiniteModulusPageFull12}{\hidepaste}
\tab{5}\spadcommand{r := (random()$GF4096) ** 20\free{GF4096x }\bound{r }}
\indentrel{3}\begin{verbatim}
             3              2
   (12)  (%CT  + %CT + 1)%CU  + %CT %CU
         Type: FiniteFieldExtension(FiniteField(2,4),3)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty12}
\begin{paste}{ugxProblemFiniteModulusPageEmpty12}{ugxProblemFiniteModulusPagePatch12}
\pastebutton{ugxProblemFiniteModulusPageEmpty12}{\showpaste}
\tab{5}\spadcommand{r := (random()$GF4096) ** 20\free{GF4096x }\bound{r }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch13}
\begin{paste}{ugxProblemFiniteModulusPageFull13}{ugxProblemFiniteModulusPageEmpty13}
\pastebutton{ugxProblemFiniteModulusPageFull13}{\hidepaste}
\tab{5}\spadcommand{norm(r)\free{r }}
\indentrel{3}\begin{verbatim}
            2
   (13)  %CT  + %CT
                                 Type: FiniteField(2,4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty13}
\begin{paste}{ugxProblemFiniteModulusPageEmpty13}{ugxProblemFiniteModulusPagePatch13}
\pastebutton{ugxProblemFiniteModulusPageEmpty13}{\showpaste}
\tab{5}\spadcommand{norm(r)\free{r }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch14}
\begin{paste}{ugxProblemFiniteModulusPageFull14}{ugxProblemFiniteModulusPageEmpty14}
\pastebutton{ugxProblemFiniteModulusPageFull14}{\hidepaste}
\tab{5}\spadcommand{GF4 := FF(2,2);\bound{GF4 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty14}
\begin{paste}{ugxProblemFiniteModulusPageEmpty14}{ugxProblemFiniteModulusPagePatch14}
\pastebutton{ugxProblemFiniteModulusPageEmpty14}{\showpaste}
\tab{5}\spadcommand{GF4 := FF(2,2);\bound{GF4 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch15}
\begin{paste}{ugxProblemFiniteModulusPageFull15}{ugxProblemFiniteModulusPageEmpty15}
\pastebutton{ugxProblemFiniteModulusPageFull15}{\hidepaste}
\tab{5}\spadcommand{f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)\free{GF4 }\bound{f }}
\indentrel{3}\begin{verbatim}
          6    5             4             2
   (15)  ?  + ?  + (%CV + 1)?  + (%CV + 1)?  + ? + %CV
Type: Union(SparseUnivariatePolynomial FiniteField(2,2),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty15}
\begin{paste}{ugxProblemFiniteModulusPageEmpty15}{ugxProblemFiniteModulusPagePatch15}
\pastebutton{ugxProblemFiniteModulusPageEmpty15}{\showpaste}
\tab{5}\spadcommand{f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)\free{GF4 }\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch16}
\begin{paste}{ugxProblemFiniteModulusPageFull16}{ugxProblemFiniteModulusPageEmpty16}
\pastebutton{ugxProblemFiniteModulusPageFull16}{\hidepaste}
\tab{5}\spadcommand{GF4096 := FFP(GF4,f);\bound{GF4096y }\free{f GF4 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty16}
\begin{paste}{ugxProblemFiniteModulusPageEmpty16}{ugxProblemFiniteModulusPagePatch16}
\pastebutton{ugxProblemFiniteModulusPageEmpty16}{\showpaste}
\tab{5}\spadcommand{GF4096 := FFP(GF4,f);\bound{GF4096y }\free{f GF4 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPagePatch17}
\begin{paste}{ugxProblemFiniteModulusPageFull17}{ugxProblemFiniteModulusPageEmpty17}
\pastebutton{ugxProblemFiniteModulusPageFull17}{\hidepaste}
\tab{5}\spadcommand{discreteLog random()$GF4096\free{GF4096y }}
\indentrel{3}\begin{verbatim}
   (17)  3370
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteModulusPageEmpty17}
\begin{paste}{ugxProblemFiniteModulusPageEmpty17}{ugxProblemFiniteModulusPagePatch17}
\pastebutton{ugxProblemFiniteModulusPageEmpty17}{\showpaste}
\tab{5}\spadcommand{discreteLog random()$GF4096\free{GF4096y }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch1}
\begin{paste}{ugProblemIdealPageFull1}{ugProblemIdealPageEmpty1}
\pastebutton{ugProblemIdealPageFull1}{\hidepaste}
\tab{5}\spadcommand{(n,m) : List DMP([x,y],FRAC INT)\bound{nm }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty1}
\begin{paste}{ugProblemIdealPageEmpty1}{ugProblemIdealPagePatch1}
\pastebutton{ugProblemIdealPageEmpty1}{\showpaste}
\tab{5}\spadcommand{(n,m) : List DMP([x,y],FRAC INT)\bound{nm }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch2}
\begin{paste}{ugProblemIdealPageFull2}{ugProblemIdealPageEmpty2}
\pastebutton{ugProblemIdealPageFull2}{\hidepaste}
\tab{5}\spadcommand{m := [x**2+y**2-1]\free{nm }\bound{m }}
\indentrel{3}\begin{verbatim}
          2    2
   (2)  [x  + y  - 1]
Type: List DistributedMultivariatePolynomial([x,y],Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty2}
\begin{paste}{ugProblemIdealPageEmpty2}{ugProblemIdealPagePatch2}
\pastebutton{ugProblemIdealPageEmpty2}{\showpaste}
\tab{5}\spadcommand{m := [x**2+y**2-1]\free{nm }\bound{m }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch3}
\begin{paste}{ugProblemIdealPageFull3}{ugProblemIdealPageEmpty3}
\pastebutton{ugProblemIdealPageFull3}{\hidepaste}
\tab{5}\spadcommand{n := [x**2-y**2]\free{nm }\bound{n }}
\indentrel{3}\begin{verbatim}
          2    2
   (3)  [x  - y ]
Type: List DistributedMultivariatePolynomial([x,y],Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty3}
\begin{paste}{ugProblemIdealPageEmpty3}{ugProblemIdealPagePatch3}
\pastebutton{ugProblemIdealPageEmpty3}{\showpaste}
\tab{5}\spadcommand{n := [x**2-y**2]\free{nm }\bound{n }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch4}
\begin{paste}{ugProblemIdealPageFull4}{ugProblemIdealPageEmpty4}
\pastebutton{ugProblemIdealPageFull4}{\hidepaste}
\tab{5}\spadcommand{id := ideal m + ideal n\free{n m }\bound{id }}
\indentrel{3}\begin{verbatim}
          2   1  2   1
   (4)  [x  - Ä,y  - Ä]
              2      2
Type: PolynomialIdeals(Fraction Integer,DirectProduct(2,NonNegativeInteger),OrderedVariableList [x,y],DistributedMultivariatePolynomial([x,y],Fraction Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty4}
\begin{paste}{ugProblemIdealPageEmpty4}{ugProblemIdealPagePatch4}
\pastebutton{ugProblemIdealPageEmpty4}{\showpaste}
\tab{5}\spadcommand{id := ideal m + ideal n\free{n m }\bound{id }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch5}
\begin{paste}{ugProblemIdealPageFull5}{ugProblemIdealPageEmpty5}
\pastebutton{ugProblemIdealPageFull5}{\hidepaste}
\tab{5}\spadcommand{zeroDim? id\free{id }}
\indentrel{3}\begin{verbatim}
   (5)  true
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty5}
\begin{paste}{ugProblemIdealPageEmpty5}{ugProblemIdealPagePatch5}
\pastebutton{ugProblemIdealPageEmpty5}{\showpaste}
\tab{5}\spadcommand{zeroDim? id\free{id }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch6}
\begin{paste}{ugProblemIdealPageFull6}{ugProblemIdealPageEmpty6}
\pastebutton{ugProblemIdealPageFull6}{\hidepaste}
\tab{5}\spadcommand{zeroDim?(ideal m)\free{m }}
\indentrel{3}\begin{verbatim}
   (6)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty6}
\begin{paste}{ugProblemIdealPageEmpty6}{ugProblemIdealPagePatch6}
\pastebutton{ugProblemIdealPageEmpty6}{\showpaste}
\tab{5}\spadcommand{zeroDim?(ideal m)\free{m }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch7}
\begin{paste}{ugProblemIdealPageFull7}{ugProblemIdealPageEmpty7}
\pastebutton{ugProblemIdealPageFull7}{\hidepaste}
\tab{5}\spadcommand{dimension ideal m\free{m }}
\indentrel{3}\begin{verbatim}
   (7)  1
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty7}
\begin{paste}{ugProblemIdealPageEmpty7}{ugProblemIdealPagePatch7}
\pastebutton{ugProblemIdealPageEmpty7}{\showpaste}
\tab{5}\spadcommand{dimension ideal m\free{m }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch8}
\begin{paste}{ugProblemIdealPageFull8}{ugProblemIdealPageEmpty8}
\pastebutton{ugProblemIdealPageFull8}{\hidepaste}
\tab{5}\spadcommand{(f,g):DMP([x,y],FRAC INT)\bound{fg }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty8}
\begin{paste}{ugProblemIdealPageEmpty8}{ugProblemIdealPagePatch8}
\pastebutton{ugProblemIdealPageEmpty8}{\showpaste}
\tab{5}\spadcommand{(f,g):DMP([x,y],FRAC INT)\bound{fg }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch9}
\begin{paste}{ugProblemIdealPageFull9}{ugProblemIdealPageEmpty9}
\pastebutton{ugProblemIdealPageFull9}{\hidepaste}
\tab{5}\spadcommand{f := x**2-1\free{fg }\bound{f }}
\indentrel{3}\begin{verbatim}
         2
   (9)  x  - 1
Type: DistributedMultivariatePolynomial([x,y],Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty9}
\begin{paste}{ugProblemIdealPageEmpty9}{ugProblemIdealPagePatch9}
\pastebutton{ugProblemIdealPageEmpty9}{\showpaste}
\tab{5}\spadcommand{f := x**2-1\free{fg }\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch10}
\begin{paste}{ugProblemIdealPageFull10}{ugProblemIdealPageEmpty10}
\pastebutton{ugProblemIdealPageFull10}{\hidepaste}
\tab{5}\spadcommand{g := x*(x**2-1)\free{fg }\bound{g }}
\indentrel{3}\begin{verbatim}
          3
   (10)  x  - x
Type: DistributedMultivariatePolynomial([x,y],Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty10}
\begin{paste}{ugProblemIdealPageEmpty10}{ugProblemIdealPagePatch10}
\pastebutton{ugProblemIdealPageEmpty10}{\showpaste}
\tab{5}\spadcommand{g := x*(x**2-1)\free{fg }\bound{g }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch11}
\begin{paste}{ugProblemIdealPageFull11}{ugProblemIdealPageEmpty11}
\pastebutton{ugProblemIdealPageFull11}{\hidepaste}
\tab{5}\spadcommand{relationsIdeal [f,g]\free{f g }}
\indentrel{3}\begin{verbatim}
   (11)
         2      3      2           2           3
   [- %CY  + %CX  + %CX ] | [%CX= x  - 1,%CY= x  - x]
Type: SuchThat(List Polynomial Fraction Integer,List Equation Polynomial Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty11}
\begin{paste}{ugProblemIdealPageEmpty11}{ugProblemIdealPagePatch11}
\pastebutton{ugProblemIdealPageEmpty11}{\showpaste}
\tab{5}\spadcommand{relationsIdeal [f,g]\free{f g }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch12}
\begin{paste}{ugProblemIdealPageFull12}{ugProblemIdealPageEmpty12}
\pastebutton{ugProblemIdealPageFull12}{\hidepaste}
\tab{5}\spadcommand{l: List DMP([x,y,z],FRAC INT)\bound{ll }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty12}
\begin{paste}{ugProblemIdealPageEmpty12}{ugProblemIdealPagePatch12}
\pastebutton{ugProblemIdealPageEmpty12}{\showpaste}
\tab{5}\spadcommand{l: List DMP([x,y,z],FRAC INT)\bound{ll }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch13}
\begin{paste}{ugProblemIdealPageFull13}{ugProblemIdealPageEmpty13}
\pastebutton{ugProblemIdealPageFull13}{\hidepaste}
\tab{5}\spadcommand{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4]\free{ll }\bound{l }}
\indentrel{3}\begin{verbatim}
           2     2    2        2
   (13)  [x  + 2y ,x z  - y z,z  - 4]
Type: List DistributedMultivariatePolynomial([x,y,z],Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty13}
\begin{paste}{ugProblemIdealPageEmpty13}{ugProblemIdealPagePatch13}
\pastebutton{ugProblemIdealPageEmpty13}{\showpaste}
\tab{5}\spadcommand{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4]\free{ll }\bound{l }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch14}
\begin{paste}{ugProblemIdealPageFull14}{ugProblemIdealPageEmpty14}
\pastebutton{ugProblemIdealPageFull14}{\hidepaste}
\tab{5}\spadcommand{ld:=primaryDecomp ideal l\free{l }\bound{ld }}
\indentrel{3}\begin{verbatim}
               1    2             1    2
   (14)  [[x + Ä y,y ,z + 2],[x - Ä y,y ,z - 2]]
               2                  2
Type: List PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty14}
\begin{paste}{ugProblemIdealPageEmpty14}{ugProblemIdealPagePatch14}
\pastebutton{ugProblemIdealPageEmpty14}{\showpaste}
\tab{5}\spadcommand{ld:=primaryDecomp ideal l\free{l }\bound{ld }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch15}
\begin{paste}{ugProblemIdealPageFull15}{ugProblemIdealPageEmpty15}
\pastebutton{ugProblemIdealPageFull15}{\hidepaste}
\tab{5}\spadcommand{reduce(intersect,ld)\free{ld }}
\indentrel{3}\begin{verbatim}
              1      2  2
   (15)  [x - Ä y z,y ,z  - 4]
              4
Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty15}
\begin{paste}{ugProblemIdealPageEmpty15}{ugProblemIdealPagePatch15}
\pastebutton{ugProblemIdealPageEmpty15}{\showpaste}
\tab{5}\spadcommand{reduce(intersect,ld)\free{ld }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch16}
\begin{paste}{ugProblemIdealPageFull16}{ugProblemIdealPageEmpty16}
\pastebutton{ugProblemIdealPageFull16}{\hidepaste}
\tab{5}\spadcommand{reduce(intersect,[radical ld.i for i in 1..2])\free{ld }}
\indentrel{3}\begin{verbatim}
               2
   (16)  [x,y,z  - 4]
Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty16}
\begin{paste}{ugProblemIdealPageEmpty16}{ugProblemIdealPagePatch16}
\pastebutton{ugProblemIdealPageEmpty16}{\showpaste}
\tab{5}\spadcommand{reduce(intersect,[radical ld.i for i in 1..2])\free{ld }}
\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPagePatch17}
\begin{paste}{ugProblemIdealPageFull17}{ugProblemIdealPageEmpty17}
\pastebutton{ugProblemIdealPageFull17}{\hidepaste}
\tab{5}\spadcommand{radical ideal l\free{l }}
\indentrel{3}\begin{verbatim}
               2
   (17)  [x,y,z  - 4]
Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemIdealPageEmpty17}
\begin{paste}{ugProblemIdealPageEmpty17}{ugProblemIdealPagePatch17}
\pastebutton{ugProblemIdealPageEmpty17}{\showpaste}
\tab{5}\spadcommand{radical ideal l\free{l }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch1}
\begin{paste}{ugxProblemFiniteUtilityPageFull1}{ugxProblemFiniteUtilityPageEmpty1}
\pastebutton{ugxProblemFiniteUtilityPageFull1}{\hidepaste}
\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty1}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty1}{ugxProblemFiniteUtilityPagePatch1}
\pastebutton{ugxProblemFiniteUtilityPageEmpty1}{\showpaste}
\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch2}
\begin{paste}{ugxProblemFiniteUtilityPageFull2}{ugxProblemFiniteUtilityPageEmpty2}
\pastebutton{ugxProblemFiniteUtilityPageFull2}{\hidepaste}
\tab{5}\spadcommand{f := createIrreduciblePoly(8)$FFPOLY(GF5)\bound{f }\free{GF5 }}
\indentrel{3}\begin{verbatim}
         8    4
   (2)  ?  + ?  + 2
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty2}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty2}{ugxProblemFiniteUtilityPagePatch2}
\pastebutton{ugxProblemFiniteUtilityPageEmpty2}{\showpaste}
\tab{5}\spadcommand{f := createIrreduciblePoly(8)$FFPOLY(GF5)\bound{f }\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch3}
\begin{paste}{ugxProblemFiniteUtilityPageFull3}{ugxProblemFiniteUtilityPageEmpty3}
\pastebutton{ugxProblemFiniteUtilityPageFull3}{\hidepaste}
\tab{5}\spadcommand{primitive?(f)$FFPOLY(GF5)\free{f }}
\indentrel{3}\begin{verbatim}
   (3)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty3}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty3}{ugxProblemFiniteUtilityPagePatch3}
\pastebutton{ugxProblemFiniteUtilityPageEmpty3}{\showpaste}
\tab{5}\spadcommand{primitive?(f)$FFPOLY(GF5)\free{f }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch4}
\begin{paste}{ugxProblemFiniteUtilityPageFull4}{ugxProblemFiniteUtilityPageEmpty4}
\pastebutton{ugxProblemFiniteUtilityPageFull4}{\hidepaste}
\tab{5}\spadcommand{normal?(f)$FFPOLY(GF5)\free{f }}
\indentrel{3}\begin{verbatim}
   (4)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty4}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty4}{ugxProblemFiniteUtilityPagePatch4}
\pastebutton{ugxProblemFiniteUtilityPageEmpty4}{\showpaste}
\tab{5}\spadcommand{normal?(f)$FFPOLY(GF5)\free{f }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch5}
\begin{paste}{ugxProblemFiniteUtilityPageFull5}{ugxProblemFiniteUtilityPageEmpty5}
\pastebutton{ugxProblemFiniteUtilityPageFull5}{\hidepaste}
\tab{5}\spadcommand{p := createPrimitivePoly(8)$FFPOLY(GF5)\bound{p }\free{GF5 }}
\indentrel{3}\begin{verbatim}
         8    3    2
   (5)  ?  + ?  + ?  + ? + 2
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty5}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty5}{ugxProblemFiniteUtilityPagePatch5}
\pastebutton{ugxProblemFiniteUtilityPageEmpty5}{\showpaste}
\tab{5}\spadcommand{p := createPrimitivePoly(8)$FFPOLY(GF5)\bound{p }\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch6}
\begin{paste}{ugxProblemFiniteUtilityPageFull6}{ugxProblemFiniteUtilityPageEmpty6}
\pastebutton{ugxProblemFiniteUtilityPageFull6}{\hidepaste}
\tab{5}\spadcommand{primitive?(p)$FFPOLY(GF5)\free{p }}
\indentrel{3}\begin{verbatim}
   (6)  true
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty6}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty6}{ugxProblemFiniteUtilityPagePatch6}
\pastebutton{ugxProblemFiniteUtilityPageEmpty6}{\showpaste}
\tab{5}\spadcommand{primitive?(p)$FFPOLY(GF5)\free{p }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch7}
\begin{paste}{ugxProblemFiniteUtilityPageFull7}{ugxProblemFiniteUtilityPageEmpty7}
\pastebutton{ugxProblemFiniteUtilityPageFull7}{\hidepaste}
\tab{5}\spadcommand{normal?(p)$FFPOLY(GF5)\free{p }}
\indentrel{3}\begin{verbatim}
   (7)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty7}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty7}{ugxProblemFiniteUtilityPagePatch7}
\pastebutton{ugxProblemFiniteUtilityPageEmpty7}{\showpaste}
\tab{5}\spadcommand{normal?(p)$FFPOLY(GF5)\free{p }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch8}
\begin{paste}{ugxProblemFiniteUtilityPageFull8}{ugxProblemFiniteUtilityPageEmpty8}
\pastebutton{ugxProblemFiniteUtilityPageFull8}{\hidepaste}
\tab{5}\spadcommand{n := createNormalPoly(8)$FFPOLY(GF5)\bound{n }\free{GF5 }}
\indentrel{3}\begin{verbatim}
         8     7    3
   (8)  ?  + 4?  + ?  + 1
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty8}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty8}{ugxProblemFiniteUtilityPagePatch8}
\pastebutton{ugxProblemFiniteUtilityPageEmpty8}{\showpaste}
\tab{5}\spadcommand{n := createNormalPoly(8)$FFPOLY(GF5)\bound{n }\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch9}
\begin{paste}{ugxProblemFiniteUtilityPageFull9}{ugxProblemFiniteUtilityPageEmpty9}
\pastebutton{ugxProblemFiniteUtilityPageFull9}{\hidepaste}
\tab{5}\spadcommand{primitive?(n)$FFPOLY(GF5)\free{n }}
\indentrel{3}\begin{verbatim}
   (9)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty9}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty9}{ugxProblemFiniteUtilityPagePatch9}
\pastebutton{ugxProblemFiniteUtilityPageEmpty9}{\showpaste}
\tab{5}\spadcommand{primitive?(n)$FFPOLY(GF5)\free{n }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch10}
\begin{paste}{ugxProblemFiniteUtilityPageFull10}{ugxProblemFiniteUtilityPageEmpty10}
\pastebutton{ugxProblemFiniteUtilityPageFull10}{\hidepaste}
\tab{5}\spadcommand{createPrimitiveNormalPoly(8)$FFPOLY(GF5)\free{GF5 }}
\indentrel{3}\begin{verbatim}
          8     7     5
   (10)  ?  + 4?  + 2?  + 2
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty10}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty10}{ugxProblemFiniteUtilityPagePatch10}
\pastebutton{ugxProblemFiniteUtilityPageEmpty10}{\showpaste}
\tab{5}\spadcommand{createPrimitiveNormalPoly(8)$FFPOLY(GF5)\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch11}
\begin{paste}{ugxProblemFiniteUtilityPageFull11}{ugxProblemFiniteUtilityPageEmpty11}
\pastebutton{ugxProblemFiniteUtilityPageFull11}{\hidepaste}
\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty11}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty11}{ugxProblemFiniteUtilityPagePatch11}
\pastebutton{ugxProblemFiniteUtilityPageEmpty11}{\showpaste}
\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch12}
\begin{paste}{ugxProblemFiniteUtilityPageFull12}{ugxProblemFiniteUtilityPageEmpty12}
\pastebutton{ugxProblemFiniteUtilityPageFull12}{\hidepaste}
\tab{5}\spadcommand{h := monomial(1,8)$SUP(GF5)\bound{h }\free{GF5 }}
\indentrel{3}\begin{verbatim}
          8
   (12)  ?
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty12}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty12}{ugxProblemFiniteUtilityPagePatch12}
\pastebutton{ugxProblemFiniteUtilityPageEmpty12}{\showpaste}
\tab{5}\spadcommand{h := monomial(1,8)$SUP(GF5)\bound{h }\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch13}
\begin{paste}{ugxProblemFiniteUtilityPageFull13}{ugxProblemFiniteUtilityPageEmpty13}
\pastebutton{ugxProblemFiniteUtilityPageFull13}{\hidepaste}
\tab{5}\spadcommand{nh := nextIrreduciblePoly(h)$FFPOLY(GF5)\bound{nh }\free{h }}
\indentrel{3}\begin{verbatim}
          8
   (13)  ?  + 2
Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty13}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty13}{ugxProblemFiniteUtilityPagePatch13}
\pastebutton{ugxProblemFiniteUtilityPageEmpty13}{\showpaste}
\tab{5}\spadcommand{nh := nextIrreduciblePoly(h)$FFPOLY(GF5)\bound{nh }\free{h }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch14}
\begin{paste}{ugxProblemFiniteUtilityPageFull14}{ugxProblemFiniteUtilityPageEmpty14}
\pastebutton{ugxProblemFiniteUtilityPageFull14}{\hidepaste}
\tab{5}\spadcommand{createIrreduciblePoly(3)$FFPOLY(GF5)\free{GF5 }}
\indentrel{3}\begin{verbatim}
          3
   (14)  ?  + ? + 1
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty14}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty14}{ugxProblemFiniteUtilityPagePatch14}
\pastebutton{ugxProblemFiniteUtilityPageEmpty14}{\showpaste}
\tab{5}\spadcommand{createIrreduciblePoly(3)$FFPOLY(GF5)\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch15}
\begin{paste}{ugxProblemFiniteUtilityPageFull15}{ugxProblemFiniteUtilityPageEmpty15}
\pastebutton{ugxProblemFiniteUtilityPageFull15}{\hidepaste}
\tab{5}\spadcommand{nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)\free{nh }}
\indentrel{3}\begin{verbatim}
          8
   (15)  ?  + 3
Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty15}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty15}{ugxProblemFiniteUtilityPagePatch15}
\pastebutton{ugxProblemFiniteUtilityPageEmpty15}{\showpaste}
\tab{5}\spadcommand{nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)\free{nh }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch16}
\begin{paste}{ugxProblemFiniteUtilityPageFull16}{ugxProblemFiniteUtilityPageEmpty16}
\pastebutton{ugxProblemFiniteUtilityPageFull16}{\hidepaste}
\tab{5}\spadcommand{numberOfIrreduciblePoly(5)$FFPOLY(GF5)\free{GF5 }}
\indentrel{3}\begin{verbatim}
   (16)  624
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty16}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty16}{ugxProblemFiniteUtilityPagePatch16}
\pastebutton{ugxProblemFiniteUtilityPageEmpty16}{\showpaste}
\tab{5}\spadcommand{numberOfIrreduciblePoly(5)$FFPOLY(GF5)\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch17}
\begin{paste}{ugxProblemFiniteUtilityPageFull17}{ugxProblemFiniteUtilityPageEmpty17}
\pastebutton{ugxProblemFiniteUtilityPageFull17}{\hidepaste}
\tab{5}\spadcommand{numberOfPrimitivePoly(3)$FFPOLY(GF5)\free{GF5 }}
\indentrel{3}\begin{verbatim}
   (17)  20
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty17}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty17}{ugxProblemFiniteUtilityPagePatch17}
\pastebutton{ugxProblemFiniteUtilityPageEmpty17}{\showpaste}
\tab{5}\spadcommand{numberOfPrimitivePoly(3)$FFPOLY(GF5)\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch18}
\begin{paste}{ugxProblemFiniteUtilityPageFull18}{ugxProblemFiniteUtilityPageEmpty18}
\pastebutton{ugxProblemFiniteUtilityPageFull18}{\hidepaste}
\tab{5}\spadcommand{m := monomial(1,1)$SUP(GF5)\bound{m }\free{GF5 }}
\indentrel{3}\begin{verbatim}
   (18)  ?
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty18}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty18}{ugxProblemFiniteUtilityPagePatch18}
\pastebutton{ugxProblemFiniteUtilityPageEmpty18}{\showpaste}
\tab{5}\spadcommand{m := monomial(1,1)$SUP(GF5)\bound{m }\free{GF5 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch19}
\begin{paste}{ugxProblemFiniteUtilityPageFull19}{ugxProblemFiniteUtilityPageEmpty19}
\pastebutton{ugxProblemFiniteUtilityPageFull19}{\hidepaste}
\tab{5}\spadcommand{f := m**3 + 4*m**2 + m + 2\bound{fx }\free{m }}
\indentrel{3}\begin{verbatim}
          3     2
   (19)  ?  + 4?  + ? + 2
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty19}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty19}{ugxProblemFiniteUtilityPagePatch19}
\pastebutton{ugxProblemFiniteUtilityPageEmpty19}{\showpaste}
\tab{5}\spadcommand{f := m**3 + 4*m**2 + m + 2\bound{fx }\free{m }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch20}
\begin{paste}{ugxProblemFiniteUtilityPageFull20}{ugxProblemFiniteUtilityPageEmpty20}
\pastebutton{ugxProblemFiniteUtilityPageFull20}{\hidepaste}
\tab{5}\spadcommand{f1 := nextPrimitivePoly(f)$FFPOLY(GF5)\free{fx }\bound{f1 }}
\indentrel{3}\begin{verbatim}
          3     2
   (20)  ?  + 4?  + 4? + 2
Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty20}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty20}{ugxProblemFiniteUtilityPagePatch20}
\pastebutton{ugxProblemFiniteUtilityPageEmpty20}{\showpaste}
\tab{5}\spadcommand{f1 := nextPrimitivePoly(f)$FFPOLY(GF5)\free{fx }\bound{f1 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch21}
\begin{paste}{ugxProblemFiniteUtilityPageFull21}{ugxProblemFiniteUtilityPageEmpty21}
\pastebutton{ugxProblemFiniteUtilityPageFull21}{\hidepaste}
\tab{5}\spadcommand{nextPrimitivePoly(f1)$FFPOLY(GF5)\free{f1 }}
\indentrel{3}\begin{verbatim}
          3     2
   (21)  ?  + 2?  + 3
Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty21}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty21}{ugxProblemFiniteUtilityPagePatch21}
\pastebutton{ugxProblemFiniteUtilityPageEmpty21}{\showpaste}
\tab{5}\spadcommand{nextPrimitivePoly(f1)$FFPOLY(GF5)\free{f1 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch22}
\begin{paste}{ugxProblemFiniteUtilityPageFull22}{ugxProblemFiniteUtilityPageEmpty22}
\pastebutton{ugxProblemFiniteUtilityPageFull22}{\hidepaste}
\tab{5}\spadcommand{f := m**3 + m**2 + 4*m + 1\bound{fy }\free{m }}
\indentrel{3}\begin{verbatim}
          3    2
   (22)  ?  + ?  + 4? + 1
          Type: SparseUnivariatePolynomial PrimeField 5
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty22}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty22}{ugxProblemFiniteUtilityPagePatch22}
\pastebutton{ugxProblemFiniteUtilityPageEmpty22}{\showpaste}
\tab{5}\spadcommand{f := m**3 + m**2 + 4*m + 1\bound{fy }\free{m }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch23}
\begin{paste}{ugxProblemFiniteUtilityPageFull23}{ugxProblemFiniteUtilityPageEmpty23}
\pastebutton{ugxProblemFiniteUtilityPageFull23}{\hidepaste}
\tab{5}\spadcommand{f1 := nextNormalPoly(f)$FFPOLY(GF5)\free{fy }\bound{f1y }}
\indentrel{3}\begin{verbatim}
          3    2
   (23)  ?  + ?  + 4? + 3
Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty23}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty23}{ugxProblemFiniteUtilityPagePatch23}
\pastebutton{ugxProblemFiniteUtilityPageEmpty23}{\showpaste}
\tab{5}\spadcommand{f1 := nextNormalPoly(f)$FFPOLY(GF5)\free{fy }\bound{f1y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch24}
\begin{paste}{ugxProblemFiniteUtilityPageFull24}{ugxProblemFiniteUtilityPageEmpty24}
\pastebutton{ugxProblemFiniteUtilityPageFull24}{\hidepaste}
\tab{5}\spadcommand{nextNormalPoly(f1)$FFPOLY(GF5)\free{f1y }}
\indentrel{3}\begin{verbatim}
          3     2
   (24)  ?  + 2?  + 1
Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty24}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty24}{ugxProblemFiniteUtilityPagePatch24}
\pastebutton{ugxProblemFiniteUtilityPageEmpty24}{\showpaste}
\tab{5}\spadcommand{nextNormalPoly(f1)$FFPOLY(GF5)\free{f1y }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch25}
\begin{paste}{ugxProblemFiniteUtilityPageFull25}{ugxProblemFiniteUtilityPageEmpty25}
\pastebutton{ugxProblemFiniteUtilityPageFull25}{\hidepaste}
\tab{5}\spadcommand{GF16 := FFX(FFX(PF 2,2),2);\bound{GF16 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty25}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty25}{ugxProblemFiniteUtilityPagePatch25}
\pastebutton{ugxProblemFiniteUtilityPageEmpty25}{\showpaste}
\tab{5}\spadcommand{GF16 := FFX(FFX(PF 2,2),2);\bound{GF16 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch26}
\begin{paste}{ugxProblemFiniteUtilityPageFull26}{ugxProblemFiniteUtilityPageEmpty26}
\pastebutton{ugxProblemFiniteUtilityPageFull26}{\hidepaste}
\tab{5}\spadcommand{createIrreduciblePoly(5)$FFPOLY(GF16)\free{GF16 }}
\indentrel{3}\begin{verbatim}
          5
   (26)  ?  + %CZ
Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty26}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty26}{ugxProblemFiniteUtilityPagePatch26}
\pastebutton{ugxProblemFiniteUtilityPageEmpty26}{\showpaste}
\tab{5}\spadcommand{createIrreduciblePoly(5)$FFPOLY(GF16)\free{GF16 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch27}
\begin{paste}{ugxProblemFiniteUtilityPageFull27}{ugxProblemFiniteUtilityPageEmpty27}
\pastebutton{ugxProblemFiniteUtilityPageFull27}{\hidepaste}
\tab{5}\spadcommand{random(5)$FFPOLY(GF16)\free{GF16 }}
\indentrel{3}\begin{verbatim}
   (27)
      5             4                            3
     ?  + (%CV + 1)?  + ((%CV + 1)%CZ + %CV + 1)?
   + 
               2
     (%CV + 1)?  + ((%CV + 1)%CZ + %CV + 1)?
   + 
     (%CV + 1)%CZ + 1
Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty27}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty27}{ugxProblemFiniteUtilityPagePatch27}
\pastebutton{ugxProblemFiniteUtilityPageEmpty27}{\showpaste}
\tab{5}\spadcommand{random(5)$FFPOLY(GF16)\free{GF16 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch28}
\begin{paste}{ugxProblemFiniteUtilityPageFull28}{ugxProblemFiniteUtilityPageEmpty28}
\pastebutton{ugxProblemFiniteUtilityPageFull28}{\hidepaste}
\tab{5}\spadcommand{random(3,9)$FFPOLY(GF16)\free{GF16 }}
\indentrel{3}\begin{verbatim}
   (28)
      4                   3                       2
     ?  + (%CZ + %CV + 1)?  + (%CV %CZ + %CV + 1)?
   + 
     (%CZ + %CV)? + (%CV + 1)%CZ
Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty28}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty28}{ugxProblemFiniteUtilityPagePatch28}
\pastebutton{ugxProblemFiniteUtilityPageEmpty28}{\showpaste}
\tab{5}\spadcommand{random(3,9)$FFPOLY(GF16)\free{GF16 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch29}
\begin{paste}{ugxProblemFiniteUtilityPageFull29}{ugxProblemFiniteUtilityPageEmpty29}
\pastebutton{ugxProblemFiniteUtilityPageFull29}{\hidepaste}
\tab{5}\spadcommand{GF2 := PrimeField 2;\bound{GF2 }}
\indentrel{3}\begin{verbatim}
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty29}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty29}{ugxProblemFiniteUtilityPagePatch29}
\pastebutton{ugxProblemFiniteUtilityPageEmpty29}{\showpaste}
\tab{5}\spadcommand{GF2 := PrimeField 2;\bound{GF2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch30}
\begin{paste}{ugxProblemFiniteUtilityPageFull30}{ugxProblemFiniteUtilityPageEmpty30}
\pastebutton{ugxProblemFiniteUtilityPageFull30}{\hidepaste}
\tab{5}\spadcommand{F := FFX(GF2,12)\bound{F }\free{GF2 }}
\indentrel{3}\begin{verbatim}
   (30)  FiniteFieldExtension(PrimeField 2,12)
                                           Type: Domain
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty30}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty30}{ugxProblemFiniteUtilityPagePatch30}
\pastebutton{ugxProblemFiniteUtilityPageEmpty30}{\showpaste}
\tab{5}\spadcommand{F := FFX(GF2,12)\bound{F }\free{GF2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch31}
\begin{paste}{ugxProblemFiniteUtilityPageFull31}{ugxProblemFiniteUtilityPageEmpty31}
\pastebutton{ugxProblemFiniteUtilityPageFull31}{\hidepaste}
\tab{5}\spadcommand{f := createIrreduciblePoly(6)$FFPOLY(GF2)\bound{fz }\free{GF2 }}
\indentrel{3}\begin{verbatim}
          6
   (31)  ?  + ? + 1
          Type: SparseUnivariatePolynomial PrimeField 2
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty31}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty31}{ugxProblemFiniteUtilityPagePatch31}
\pastebutton{ugxProblemFiniteUtilityPageEmpty31}{\showpaste}
\tab{5}\spadcommand{f := createIrreduciblePoly(6)$FFPOLY(GF2)\bound{fz }\free{GF2 }}
\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPagePatch32}
\begin{paste}{ugxProblemFiniteUtilityPageFull32}{ugxProblemFiniteUtilityPageEmpty32}
\pastebutton{ugxProblemFiniteUtilityPageFull32}{\hidepaste}
\tab{5}\spadcommand{root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)\free{F GF2 fz }\bound{root }}
\indentrel{3}\begin{verbatim}
            11      8      7      5
   (32)  %CS   + %CS  + %CS  + %CS  + %CS + 1
            Type: FiniteFieldExtension(PrimeField 2,12)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugxProblemFiniteUtilityPageEmpty32}
\begin{paste}{ugxProblemFiniteUtilityPageEmpty32}{ugxProblemFiniteUtilityPagePatch32}
\pastebutton{ugxProblemFiniteUtilityPageEmpty32}{\showpaste}
\tab{5}\spadcommand{root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)\free{F GF2 fz }\bound{root }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch1}
\begin{paste}{ugProblemEigenPageFull1}{ugProblemEigenPageEmpty1}
\pastebutton{ugProblemEigenPageFull1}{\hidepaste}
\tab{5}\spadcommand{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]]\bound{m1 }}
\indentrel{3}\begin{verbatim}
        Ú1   2    1 ¿
        ³           ³
   (1)  ³2   1   - 2³
        ³           ³
        À1  - 2   4 Ù
                                   Type: Matrix Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty1}
\begin{paste}{ugProblemEigenPageEmpty1}{ugProblemEigenPagePatch1}
\pastebutton{ugProblemEigenPageEmpty1}{\showpaste}
\tab{5}\spadcommand{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]]\bound{m1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch2}
\begin{paste}{ugProblemEigenPageFull2}{ugProblemEigenPageEmpty2}
\pastebutton{ugProblemEigenPageFull2}{\hidepaste}
\tab{5}\spadcommand{leig := eigenvalues(m1)\free{m1 }\bound{leig }}
\indentrel{3}\begin{verbatim}
                    2
   (2)  [5,%DA | %DA  - %DA - 5]
Type: List Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty2}
\begin{paste}{ugProblemEigenPageEmpty2}{ugProblemEigenPagePatch2}
\pastebutton{ugProblemEigenPageEmpty2}{\showpaste}
\tab{5}\spadcommand{leig := eigenvalues(m1)\free{m1 }\bound{leig }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch3}
\begin{paste}{ugProblemEigenPageFull3}{ugProblemEigenPageEmpty3}
\pastebutton{ugProblemEigenPageFull3}{\hidepaste}
\tab{5}\spadcommand{eigenvector(first(leig),m1)\free{m1 leig }}
\indentrel{3}\begin{verbatim}
         Ú 0 ¿
         ³   ³
         ³  1³
   (3)  [³- ij]
         ³  2³
         ³   ³
         À 1 Ù
 Type: List Matrix Fraction Polynomial Fraction Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty3}
\begin{paste}{ugProblemEigenPageEmpty3}{ugProblemEigenPagePatch3}
\pastebutton{ugProblemEigenPageEmpty3}{\showpaste}
\tab{5}\spadcommand{eigenvector(first(leig),m1)\free{m1 leig }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch4}
\begin{paste}{ugProblemEigenPageFull4}{ugProblemEigenPageEmpty4}
\pastebutton{ugProblemEigenPageFull4}{\hidepaste}
\tab{5}\spadcommand{eigenvectors(m1)\free{m1 }}
\indentrel{3}\begin{verbatim}
   (4)
                                   Ú 0 ¿
                                   ³   ³
                                   ³  1³
   [[eigval= 5,eigmult= 1,eigvec= [³- ij]],
                                   ³  2³
                                   ³   ³
                                   À 1 Ù

                        2
     [eigval= (%DB | %DB  - %DB - 5), eigmult= 1,
               Ú%DB¿
               ³   ³
      eigvec= [³ 2 ³]]
               ³   ³
               À 1 Ù
     ]
Type: List Record(eigval: Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)),eigmult: NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty4}
\begin{paste}{ugProblemEigenPageEmpty4}{ugProblemEigenPagePatch4}
\pastebutton{ugProblemEigenPageEmpty4}{\showpaste}
\tab{5}\spadcommand{eigenvectors(m1)\free{m1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch5}
\begin{paste}{ugProblemEigenPageFull5}{ugProblemEigenPageEmpty5}
\pastebutton{ugProblemEigenPageFull5}{\hidepaste}
\tab{5}\spadcommand{radicalEigenvectors(m1)\free{m1 }}
\indentrel{3}\begin{verbatim}
   (5)
                                            Ú ÚÄÄ¿    ¿
              ÚÄÄ¿                          ³\³21  + 1³
             \³21  + 1                      ³ÄÄÄÄÄÄÄÄij
   [[radval= ÄÄÄÄÄÄÄÄÄ,radmult= 1,radvect= [³    2    ³]],
                 2                          ³         ³
                                            ³    2    ³
                                            ³         ³
                                            À    1    Ù

                 ÚÄÄ¿
              - \³21  + 1
     [radval= ÄÄÄÄÄÄÄÄÄÄÄ, radmult= 1,
                   2
                Ú   ÚÄÄ¿    ¿
                ³- \³21  + 1³
                ³ÄÄÄÄÄÄÄÄÄÄij
      radvect= [³     2     ³]]
                ³           ³
                ³     2     ³
                ³           ³
                À     1     Ù
     ,
                                    Ú 0 ¿
                                    ³   ³
                                    ³  1³
    [radval= 5,radmult= 1,radvect= [³- ij]]]
                                    ³  2³
                                    ³   ³
                                    À 1 Ù
Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty5}
\begin{paste}{ugProblemEigenPageEmpty5}{ugProblemEigenPagePatch5}
\pastebutton{ugProblemEigenPageEmpty5}{\showpaste}
\tab{5}\spadcommand{radicalEigenvectors(m1)\free{m1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch6}
\begin{paste}{ugProblemEigenPageFull6}{ugProblemEigenPageEmpty6}
\pastebutton{ugProblemEigenPageFull6}{\hidepaste}
\tab{5}\spadcommand{realEigenvectors(m1,1/1000)\free{m1 }}
\indentrel{3}\begin{verbatim}
   (6)
                                    Ú 0 ¿
                                    ³   ³
                                    ³  1³
   [[outval= 5,outmult= 1,outvect= [³- ij]],
                                    ³  2³
                                    ³   ³
                                    À 1 Ù
                                       Ú5717¿
                                       ³ÄÄÄij
             5717                      ³2048³
    [outval= ÄÄÄÄ,outmult= 1,outvect= [³    ³]],
             2048                      ³ 2  ³
                                       ³    ³
                                       À 1  Ù
                                         Ú  3669¿
                                         ³- ÄÄÄij
               3669                      ³  2048³
    [outval= - ÄÄÄÄ,outmult= 1,outvect= [³      ³]]]
               2048                      ³  2   ³
                                         ³      ³
                                         À  1   Ù
Type: List Record(outval: Fraction Integer,outmult: Integer,outvect: List Matrix Fraction Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty6}
\begin{paste}{ugProblemEigenPageEmpty6}{ugProblemEigenPagePatch6}
\pastebutton{ugProblemEigenPageEmpty6}{\showpaste}
\tab{5}\spadcommand{realEigenvectors(m1,1/1000)\free{m1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch7}
\begin{paste}{ugProblemEigenPageFull7}{ugProblemEigenPageEmpty7}
\pastebutton{ugProblemEigenPageFull7}{\hidepaste}
\tab{5}\spadcommand{eigenMatrix(m1)\free{m1 }}
\indentrel{3}\begin{verbatim}
        Ú ÚÄÄ¿         ÚÄÄ¿         ¿
        ³\³21  + 1  - \³21  + 1     ³
        ³ÄÄÄÄÄÄÄÄÄ  ÄÄÄÄÄÄÄÄÄÄÄ   0 ³
        ³    2           2          ³
   (7)  ³                           ³
        ³                          1³
        ³    2           2       - ij
        ³                          2³
        ³                           ³
        À    1           1        1 Ù
             Type: Union(Matrix Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty7}
\begin{paste}{ugProblemEigenPageEmpty7}{ugProblemEigenPagePatch7}
\pastebutton{ugProblemEigenPageEmpty7}{\showpaste}
\tab{5}\spadcommand{eigenMatrix(m1)\free{m1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch8}
\begin{paste}{ugProblemEigenPageFull8}{ugProblemEigenPageEmpty8}
\pastebutton{ugProblemEigenPageFull8}{\hidepaste}
\tab{5}\spadcommand{m2 := matrix [[-5,-2],[18,7]]\bound{m2 }}
\indentrel{3}\begin{verbatim}
        Ú- 5  - 2¿
   (8)  ³        ³
        À18    7 Ù
                                   Type: Matrix Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty8}
\begin{paste}{ugProblemEigenPageEmpty8}{ugProblemEigenPagePatch8}
\pastebutton{ugProblemEigenPageEmpty8}{\showpaste}
\tab{5}\spadcommand{m2 := matrix [[-5,-2],[18,7]]\bound{m2 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch9}
\begin{paste}{ugProblemEigenPageFull9}{ugProblemEigenPageEmpty9}
\pastebutton{ugProblemEigenPageFull9}{\hidepaste}
\tab{5}\spadcommand{eigenMatrix(m2)\free{m2 }}
\indentrel{3}\begin{verbatim}
   (9)  "failed"
                              Type: Union("failed",...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty9}
\begin{paste}{ugProblemEigenPageEmpty9}{ugProblemEigenPagePatch9}
\pastebutton{ugProblemEigenPageEmpty9}{\showpaste}
\tab{5}\spadcommand{eigenMatrix(m2)\free{m2 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch10}
\begin{paste}{ugProblemEigenPageFull10}{ugProblemEigenPageEmpty10}
\pastebutton{ugProblemEigenPageFull10}{\hidepaste}
\tab{5}\spadcommand{m3 := matrix [[1,2],[2,1]]\bound{m3 }}
\indentrel{3}\begin{verbatim}
         Ú1  2¿
   (10)  ³    ³
         À2  1Ù
                                   Type: Matrix Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty10}
\begin{paste}{ugProblemEigenPageEmpty10}{ugProblemEigenPagePatch10}
\pastebutton{ugProblemEigenPageEmpty10}{\showpaste}
\tab{5}\spadcommand{m3 := matrix [[1,2],[2,1]]\bound{m3 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPagePatch11}
\begin{paste}{ugProblemEigenPageFull11}{ugProblemEigenPageEmpty11}
\pastebutton{ugProblemEigenPageFull11}{\hidepaste}
\tab{5}\spadcommand{orthonormalBasis(m3)\free{m3 }}
\indentrel{3}\begin{verbatim}
          Ú    1 ¿ Ú  1 ¿
          ³- ÄÄÄij ³ÄÄÄij
          ³   ÚÄ¿³ ³ ÚÄ¿³
          ³  \³2 ³ ³\³2 ³
   (11)  [³      ³,³    ³]
          ³   1  ³ ³  1 ³
          ³ ÄÄÄÄ ³ ³ÄÄÄij
          ³  ÚÄ¿ ³ ³ ÚÄ¿³
          À \³2  Ù À\³2 Ù
                   Type: List Matrix Expression Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemEigenPageEmpty11}
\begin{paste}{ugProblemEigenPageEmpty11}{ugProblemEigenPagePatch11}
\pastebutton{ugProblemEigenPageEmpty11}{\showpaste}
\tab{5}\spadcommand{orthonormalBasis(m3)\free{m3 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch1}
\begin{paste}{ugProblemGaloisPageFull1}{ugProblemGaloisPageEmpty1}
\pastebutton{ugProblemGaloisPageFull1}{\hidepaste}
\tab{5}\spadcommand{p := x**5 - 5*x + 12\bound{p }}
\indentrel{3}\begin{verbatim}
         5
   (1)  x  - 5x + 12
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty1}
\begin{paste}{ugProblemGaloisPageEmpty1}{ugProblemGaloisPagePatch1}
\pastebutton{ugProblemGaloisPageEmpty1}{\showpaste}
\tab{5}\spadcommand{p := x**5 - 5*x + 12\bound{p }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch2}
\begin{paste}{ugProblemGaloisPageFull2}{ugProblemGaloisPageEmpty2}
\pastebutton{ugProblemGaloisPageFull2}{\hidepaste}
\tab{5}\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y)\free{p }\bound{q }}
\indentrel{3}\begin{verbatim}
   (2)
      25      21        17         15        13
     x   - 50x   - 2375x   + 90000x   - 5000x
   + 
             11          9            7            5
     2700000x   + 250000x  + 18000000x  + 64000000x
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty2}
\begin{paste}{ugProblemGaloisPageEmpty2}{ugProblemGaloisPagePatch2}
\pastebutton{ugProblemGaloisPageEmpty2}{\showpaste}
\tab{5}\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y)\free{p }\bound{q }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch3}
\begin{paste}{ugProblemGaloisPageFull3}{ugProblemGaloisPageEmpty3}
\pastebutton{ugProblemGaloisPageFull3}{\hidepaste}
\tab{5}\spadcommand{q1 := exquo(q, x**5)\free{q }\bound{q1 }}
\indentrel{3}\begin{verbatim}
   (3)
      20      16        12         10        8           6
     x   - 50x   - 2375x   + 90000x   - 5000x  + 2700000x
   + 
            4            2
     250000x  + 18000000x  + 64000000
                    Type: Union(Polynomial Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty3}
\begin{paste}{ugProblemGaloisPageEmpty3}{ugProblemGaloisPagePatch3}
\pastebutton{ugProblemGaloisPageEmpty3}{\showpaste}
\tab{5}\spadcommand{q1 := exquo(q, x**5)\free{q }\bound{q1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch4}
\begin{paste}{ugProblemGaloisPageFull4}{ugProblemGaloisPageEmpty4}
\pastebutton{ugProblemGaloisPageFull4}{\hidepaste}
\tab{5}\spadcommand{factoredQ := factor q1\free{q1 }\bound{factoredQ }}
\indentrel{3}\begin{verbatim}
   (4)
       10      8      6        4        2
     (x   - 10x  - 75x  + 1500x  - 5500x  + 16000)
  *
       10      8       6       4        2
     (x   + 10x  + 125x  + 500x  + 2500x  + 4000)
                      Type: Factored Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty4}
\begin{paste}{ugProblemGaloisPageEmpty4}{ugProblemGaloisPagePatch4}
\pastebutton{ugProblemGaloisPageEmpty4}{\showpaste}
\tab{5}\spadcommand{factoredQ := factor q1\free{q1 }\bound{factoredQ }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch5}
\begin{paste}{ugProblemGaloisPageFull5}{ugProblemGaloisPageEmpty5}
\pastebutton{ugProblemGaloisPageFull5}{\hidepaste}
\tab{5}\spadcommand{r := nthFactor(factoredQ,1)\free{factoredQ }\bound{r }}
\indentrel{3}\begin{verbatim}
         10      8      6        4        2
   (5)  x   - 10x  - 75x  + 1500x  - 5500x  + 16000
                               Type: Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty5}
\begin{paste}{ugProblemGaloisPageEmpty5}{ugProblemGaloisPagePatch5}
\pastebutton{ugProblemGaloisPageEmpty5}{\showpaste}
\tab{5}\spadcommand{r := nthFactor(factoredQ,1)\free{factoredQ }\bound{r }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch6}
\begin{paste}{ugProblemGaloisPageFull6}{ugProblemGaloisPageEmpty6}
\pastebutton{ugProblemGaloisPageFull6}{\hidepaste}
\tab{5}\spadcommand{beta:AN := rootOf(eval(r,x,b))\free{r }\bound{beta }}
\indentrel{3}\begin{verbatim}
   (6)  b
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty6}
\begin{paste}{ugProblemGaloisPageEmpty6}{ugProblemGaloisPagePatch6}
\pastebutton{ugProblemGaloisPageEmpty6}{\showpaste}
\tab{5}\spadcommand{beta:AN := rootOf(eval(r,x,b))\free{r }\bound{beta }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch7}
\begin{paste}{ugProblemGaloisPageFull7}{ugProblemGaloisPageEmpty7}
\pastebutton{ugProblemGaloisPageFull7}{\hidepaste}
\tab{5}\spadcommand{p := p::UP(x,INT)::UP(x,AN)\free{p }\bound{declareP }}
\indentrel{3}\begin{verbatim}
         5
   (7)  x  - 5x + 12
          Type: UnivariatePolynomial(x,AlgebraicNumber)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty7}
\begin{paste}{ugProblemGaloisPageEmpty7}{ugProblemGaloisPagePatch7}
\pastebutton{ugProblemGaloisPageEmpty7}{\showpaste}
\tab{5}\spadcommand{p := p::UP(x,INT)::UP(x,AN)\free{p }\bound{declareP }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch8}
\begin{paste}{ugProblemGaloisPageFull8}{ugProblemGaloisPageEmpty8}
\pastebutton{ugProblemGaloisPageFull8}{\hidepaste}
\tab{5}\spadcommand{algFactors := factor(p,[beta])\free{declareP beta }\bound{algFactors }}
\indentrel{3}\begin{verbatim}
   (8)
       x
     + 
                9       8       7        6         5
           - 85b  - 116b  + 780b  + 2640b  + 14895b
         + 
                  4          3          2
           - 8820b  - 127050b  - 327000b  - 405200b
         + 
           2062400
      /
         1339200
  *
               8       6        4         2
          - 17b  + 156b  + 2979b  - 25410b  - 14080
     (x + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
                            66960
  *
       x
     + 
               8        6         4          2
           143b  - 2100b  - 10485b  + 290550b  - 334800b
         + 
           - 960800
      /
         669600
  *
       x
     + 
               8        6         4          2
           143b  - 2100b  - 10485b  + 290550b  + 334800b
         + 
           - 960800
      /
         669600
  *
       x
     + 
              9       8       7        6         5
           85b  - 116b  - 780b  + 2640b  - 14895b
         + 
                  4          3          2
           - 8820b  + 127050b  - 327000b  + 405200b
         + 
           2062400
      /
         1339200
 Type: Factored UnivariatePolynomial(x,AlgebraicNumber)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty8}
\begin{paste}{ugProblemGaloisPageEmpty8}{ugProblemGaloisPagePatch8}
\pastebutton{ugProblemGaloisPageEmpty8}{\showpaste}
\tab{5}\spadcommand{algFactors := factor(p,[beta])\free{declareP beta }\bound{algFactors }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch9}
\begin{paste}{ugProblemGaloisPageFull9}{ugProblemGaloisPageEmpty9}
\pastebutton{ugProblemGaloisPageFull9}{\hidepaste}
\tab{5}\spadcommand{factor(p)\free{declareP }}
\indentrel{3}\begin{verbatim}
         5
   (9)  x  - 5x + 12
 Type: Factored UnivariatePolynomial(x,AlgebraicNumber)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty9}
\begin{paste}{ugProblemGaloisPageEmpty9}{ugProblemGaloisPagePatch9}
\pastebutton{ugProblemGaloisPageEmpty9}{\showpaste}
\tab{5}\spadcommand{factor(p)\free{declareP }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch10}
\begin{paste}{ugProblemGaloisPageFull10}{ugProblemGaloisPageEmpty10}
\pastebutton{ugProblemGaloisPageFull10}{\hidepaste}
\tab{5}\spadcommand{factor1 := nthFactor(algFactors,1)\free{algFactors }\bound{factor1 }}
\indentrel{3}\begin{verbatim}
   (10)
     x
   + 
              9       8       7        6         5
         - 85b  - 116b  + 780b  + 2640b  + 14895b
       + 
              4          3          2
       - 8820b  - 127050b  - 327000b  - 405200b + 2062400
    /
       1339200
          Type: UnivariatePolynomial(x,AlgebraicNumber)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty10}
\begin{paste}{ugProblemGaloisPageEmpty10}{ugProblemGaloisPagePatch10}
\pastebutton{ugProblemGaloisPageEmpty10}{\showpaste}
\tab{5}\spadcommand{factor1 := nthFactor(algFactors,1)\free{algFactors }\bound{factor1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch11}
\begin{paste}{ugProblemGaloisPageFull11}{ugProblemGaloisPageEmpty11}
\pastebutton{ugProblemGaloisPageFull11}{\hidepaste}
\tab{5}\spadcommand{root1 := -coefficient(factor1,0)\free{factor1 }\bound{root1 }}
\indentrel{3}\begin{verbatim}
   (11)
          9       8       7        6         5        4
       85b  + 116b  - 780b  - 2640b  - 14895b  + 8820b
     + 
              3          2
       127050b  + 327000b  + 405200b - 2062400
  /
     1339200
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty11}
\begin{paste}{ugProblemGaloisPageEmpty11}{ugProblemGaloisPagePatch11}
\pastebutton{ugProblemGaloisPageEmpty11}{\showpaste}
\tab{5}\spadcommand{root1 := -coefficient(factor1,0)\free{factor1 }\bound{root1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch12}
\begin{paste}{ugProblemGaloisPageFull12}{ugProblemGaloisPageEmpty12}
\pastebutton{ugProblemGaloisPageFull12}{\hidepaste}
\tab{5}\spadcommand{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5]\free{algFactors }\bound{roots }}
\indentrel{3}\begin{verbatim}
   (12)
   [
            9       8       7        6         5        4
         85b  + 116b  - 780b  - 2640b  - 14895b  + 8820b
       + 
                3          2
         127050b  + 327000b  + 405200b - 2062400
    /
       1339200
     ,
       8       6        4         2
    17b  - 156b  - 2979b  + 25410b  + 14080
    ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
                     66960

               8        6         4          2
         - 143b  + 2100b  + 10485b  - 290550b  + 334800b
       + 
         960800
    /
       669600
     ,

               8        6         4          2
         - 143b  + 2100b  + 10485b  - 290550b  - 334800b
       + 
         960800
    /
       669600
     ,

              9       8       7        6         5
         - 85b  + 116b  + 780b  - 2640b  + 14895b
       + 
              4          3          2
         8820b  - 127050b  + 327000b  - 405200b - 2062400
    /
       1339200
     ]
                             Type: List AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty12}
\begin{paste}{ugProblemGaloisPageEmpty12}{ugProblemGaloisPagePatch12}
\pastebutton{ugProblemGaloisPageEmpty12}{\showpaste}
\tab{5}\spadcommand{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5]\free{algFactors }\bound{roots }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch13}
\begin{paste}{ugProblemGaloisPageFull13}{ugProblemGaloisPageEmpty13}
\pastebutton{ugProblemGaloisPageFull13}{\hidepaste}
\tab{5}\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5)\free{roots }\bound{ais }}
\indentrel{3}\begin{verbatim}
   (13)
            9       8       7        6         5        4
       - 85b  + 116b  + 780b  - 2640b  + 14895b  + 8820b
     + 
                3          2
       - 127050b  + 327000b  - 405200b - 2062400
  /
     1339200
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty13}
\begin{paste}{ugProblemGaloisPageEmpty13}{ugProblemGaloisPagePatch13}
\pastebutton{ugProblemGaloisPageEmpty13}{\showpaste}
\tab{5}\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5)\free{roots }\bound{ais }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch14}
\begin{paste}{ugProblemGaloisPageFull14}{ugProblemGaloisPageEmpty14}
\pastebutton{ugProblemGaloisPageFull14}{\hidepaste}
\tab{5}\spadcommand{eval(r,x,a1 - a2)\free{ais }}
\indentrel{3}\begin{verbatim}
   (14)  0
                       Type: Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty14}
\begin{paste}{ugProblemGaloisPageEmpty14}{ugProblemGaloisPagePatch14}
\pastebutton{ugProblemGaloisPageEmpty14}{\showpaste}
\tab{5}\spadcommand{eval(r,x,a1 - a2)\free{ais }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch15}
\begin{paste}{ugProblemGaloisPageFull15}{ugProblemGaloisPageEmpty15}
\pastebutton{ugProblemGaloisPageFull15}{\hidepaste}
\tab{5}\spadcommand{eval(r,x,a1 - a3)\free{ais }}
\indentrel{3}\begin{verbatim}
   (15)
             9         8          7          6           5
       47905b  + 66920b  - 536100b  - 980400b  - 3345075b
     + 
                 4            3             2
       - 5787000b  + 75572250b  + 161688000b  - 184600000b
     + 
       - 710912000
  /
     4464
                       Type: Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty15}
\begin{paste}{ugProblemGaloisPageEmpty15}{ugProblemGaloisPagePatch15}
\pastebutton{ugProblemGaloisPageEmpty15}{\showpaste}
\tab{5}\spadcommand{eval(r,x,a1 - a3)\free{ais }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch16}
\begin{paste}{ugProblemGaloisPageFull16}{ugProblemGaloisPageEmpty16}
\pastebutton{ugProblemGaloisPageFull16}{\hidepaste}
\tab{5}\spadcommand{eval(r,x,a1 - a4)\free{ais }}
\indentrel{3}\begin{verbatim}
   (16)  0
                       Type: Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty16}
\begin{paste}{ugProblemGaloisPageEmpty16}{ugProblemGaloisPagePatch16}
\pastebutton{ugProblemGaloisPageEmpty16}{\showpaste}
\tab{5}\spadcommand{eval(r,x,a1 - a4)\free{ais }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch17}
\begin{paste}{ugProblemGaloisPageFull17}{ugProblemGaloisPageEmpty17}
\pastebutton{ugProblemGaloisPageFull17}{\hidepaste}
\tab{5}\spadcommand{eval(r,x,a1 - a5)\free{ais }}
\indentrel{3}\begin{verbatim}
             8        6         4          2
         405b  + 3450b  - 19875b  - 198000b  - 588000
   (17)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                              31
                       Type: Polynomial AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty17}
\begin{paste}{ugProblemGaloisPageEmpty17}{ugProblemGaloisPagePatch17}
\pastebutton{ugProblemGaloisPageEmpty17}{\showpaste}
\tab{5}\spadcommand{eval(r,x,a1 - a5)\free{ais }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch18}
\begin{paste}{ugProblemGaloisPageFull18}{ugProblemGaloisPageEmpty18}
\pastebutton{ugProblemGaloisPageFull18}{\hidepaste}
\tab{5}\spadcommand{bb := a1 - a4\free{ais }\bound{bb }}
\indentrel{3}\begin{verbatim}
   (18)
          9       8       7        6         5         4
       85b  + 402b  - 780b  - 6840b  - 14895b  - 12150b
     + 
              3          2
       127050b  + 908100b  + 1074800b - 3984000
  /
     1339200
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty18}
\begin{paste}{ugProblemGaloisPageEmpty18}{ugProblemGaloisPagePatch18}
\pastebutton{ugProblemGaloisPageEmpty18}{\showpaste}
\tab{5}\spadcommand{bb := a1 - a4\free{ais }\bound{bb }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch19}
\begin{paste}{ugProblemGaloisPageFull19}{ugProblemGaloisPageEmpty19}
\pastebutton{ugProblemGaloisPageFull19}{\hidepaste}
\tab{5}\spadcommand{aa1 := subst(a1,beta = bb)\free{beta bb ais }\bound{aa1 }}
\indentrel{3}\begin{verbatim}
   (19)
             8        6         4          2
       - 143b  + 2100b  + 10485b  - 290550b  + 334800b
     + 
       960800
  /
     669600
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty19}
\begin{paste}{ugProblemGaloisPageEmpty19}{ugProblemGaloisPagePatch19}
\pastebutton{ugProblemGaloisPageEmpty19}{\showpaste}
\tab{5}\spadcommand{aa1 := subst(a1,beta = bb)\free{beta bb ais }\bound{aa1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch20}
\begin{paste}{ugProblemGaloisPageFull20}{ugProblemGaloisPageEmpty20}
\pastebutton{ugProblemGaloisPageFull20}{\hidepaste}
\tab{5}\spadcommand{aa2 := subst(a2,beta = bb)\free{beta bb ais }\bound{aa2 }}
\indentrel{3}\begin{verbatim}
   (20)
            9       8       7        6         5        4
       - 85b  + 116b  + 780b  - 2640b  + 14895b  + 8820b
     + 
                3          2
       - 127050b  + 327000b  - 405200b - 2062400
  /
     1339200
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty20}
\begin{paste}{ugProblemGaloisPageEmpty20}{ugProblemGaloisPagePatch20}
\pastebutton{ugProblemGaloisPageEmpty20}{\showpaste}
\tab{5}\spadcommand{aa2 := subst(a2,beta = bb)\free{beta bb ais }\bound{aa2 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch21}
\begin{paste}{ugProblemGaloisPageFull21}{ugProblemGaloisPageEmpty21}
\pastebutton{ugProblemGaloisPageFull21}{\hidepaste}
\tab{5}\spadcommand{aa3 := subst(a3,beta = bb)\free{beta bb ais }\bound{aa3 }}
\indentrel{3}\begin{verbatim}
   (21)
          9       8       7        6         5        4
       85b  + 116b  - 780b  - 2640b  - 14895b  + 8820b
     + 
              3          2
       127050b  + 327000b  + 405200b - 2062400
  /
     1339200
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty21}
\begin{paste}{ugProblemGaloisPageEmpty21}{ugProblemGaloisPagePatch21}
\pastebutton{ugProblemGaloisPageEmpty21}{\showpaste}
\tab{5}\spadcommand{aa3 := subst(a3,beta = bb)\free{beta bb ais }\bound{aa3 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch22}
\begin{paste}{ugProblemGaloisPageFull22}{ugProblemGaloisPageEmpty22}
\pastebutton{ugProblemGaloisPageFull22}{\hidepaste}
\tab{5}\spadcommand{aa4 := subst(a4,beta = bb)\free{beta bb ais }\bound{aa4 }}
\indentrel{3}\begin{verbatim}
   (22)
             8        6         4          2
       - 143b  + 2100b  + 10485b  - 290550b  - 334800b
     + 
       960800
  /
     669600
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty22}
\begin{paste}{ugProblemGaloisPageEmpty22}{ugProblemGaloisPagePatch22}
\pastebutton{ugProblemGaloisPageEmpty22}{\showpaste}
\tab{5}\spadcommand{aa4 := subst(a4,beta = bb)\free{beta bb ais }\bound{aa4 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch23}
\begin{paste}{ugProblemGaloisPageFull23}{ugProblemGaloisPageEmpty23}
\pastebutton{ugProblemGaloisPageFull23}{\hidepaste}
\tab{5}\spadcommand{aa5 := subst(a5,beta = bb)\free{beta bb ais }\bound{aa5 }}
\indentrel{3}\begin{verbatim}
            8       6        4         2
         17b  - 156b  - 2979b  + 25410b  + 14080
   (23)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                          66960
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty23}
\begin{paste}{ugProblemGaloisPageEmpty23}{ugProblemGaloisPagePatch23}
\pastebutton{ugProblemGaloisPageEmpty23}{\showpaste}
\tab{5}\spadcommand{aa5 := subst(a5,beta = bb)\free{beta bb ais }\bound{aa5 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch24}
\begin{paste}{ugProblemGaloisPageFull24}{ugProblemGaloisPageEmpty24}
\pastebutton{ugProblemGaloisPageFull24}{\hidepaste}
\tab{5}\spadcommand{(aa1 = a1) :: Boolean\free{aa1 }}
\indentrel{3}\begin{verbatim}
   (24)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty24}
\begin{paste}{ugProblemGaloisPageEmpty24}{ugProblemGaloisPagePatch24}
\pastebutton{ugProblemGaloisPageEmpty24}{\showpaste}
\tab{5}\spadcommand{(aa1 = a1) :: Boolean\free{aa1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch25}
\begin{paste}{ugProblemGaloisPageFull25}{ugProblemGaloisPageEmpty25}
\pastebutton{ugProblemGaloisPageFull25}{\hidepaste}
\tab{5}\spadcommand{(aa1 = a2) :: Boolean\free{aa1 }}
\indentrel{3}\begin{verbatim}
   (25)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty25}
\begin{paste}{ugProblemGaloisPageEmpty25}{ugProblemGaloisPagePatch25}
\pastebutton{ugProblemGaloisPageEmpty25}{\showpaste}
\tab{5}\spadcommand{(aa1 = a2) :: Boolean\free{aa1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch26}
\begin{paste}{ugProblemGaloisPageFull26}{ugProblemGaloisPageEmpty26}
\pastebutton{ugProblemGaloisPageFull26}{\hidepaste}
\tab{5}\spadcommand{(aa1 = a3) :: Boolean\free{aa1 }}
\indentrel{3}\begin{verbatim}
   (26)  true
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty26}
\begin{paste}{ugProblemGaloisPageEmpty26}{ugProblemGaloisPagePatch26}
\pastebutton{ugProblemGaloisPageEmpty26}{\showpaste}
\tab{5}\spadcommand{(aa1 = a3) :: Boolean\free{aa1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch27}
\begin{paste}{ugProblemGaloisPageFull27}{ugProblemGaloisPageEmpty27}
\pastebutton{ugProblemGaloisPageFull27}{\hidepaste}
\tab{5}\spadcommand{(aa1 = a4) :: Boolean\free{aa1 }}
\indentrel{3}\begin{verbatim}
   (27)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty27}
\begin{paste}{ugProblemGaloisPageEmpty27}{ugProblemGaloisPagePatch27}
\pastebutton{ugProblemGaloisPageEmpty27}{\showpaste}
\tab{5}\spadcommand{(aa1 = a4) :: Boolean\free{aa1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPagePatch28}
\begin{paste}{ugProblemGaloisPageFull28}{ugProblemGaloisPageEmpty28}
\pastebutton{ugProblemGaloisPageFull28}{\hidepaste}
\tab{5}\spadcommand{(aa1 = a5) :: Boolean\free{aa1 }}
\indentrel{3}\begin{verbatim}
   (28)  false
                                          Type: Boolean
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemGaloisPageEmpty28}
\begin{paste}{ugProblemGaloisPageEmpty28}{ugProblemGaloisPagePatch28}
\pastebutton{ugProblemGaloisPageEmpty28}{\showpaste}
\tab{5}\spadcommand{(aa1 = a5) :: Boolean\free{aa1 }}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch1}
\begin{paste}{ugProblemLimitsPageFull1}{ugProblemLimitsPageEmpty1}
\pastebutton{ugProblemLimitsPageFull1}{\hidepaste}
\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
\indentrel{3}\begin{verbatim}
          1
   (1)  - Ä
          2
Type: Union(OrderedCompletion Fraction Polynomial Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty1}
\begin{paste}{ugProblemLimitsPageEmpty1}{ugProblemLimitsPagePatch1}
\pastebutton{ugProblemLimitsPageEmpty1}{\showpaste}
\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch2}
\begin{paste}{ugProblemLimitsPageFull2}{ugProblemLimitsPageEmpty2}
\pastebutton{ugProblemLimitsPageFull2}{\hidepaste}
\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")}
\indentrel{3}\begin{verbatim}
   (2)  0
  Type: Union(OrderedCompletion Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty2}
\begin{paste}{ugProblemLimitsPageEmpty2}{ugProblemLimitsPagePatch2}
\pastebutton{ugProblemLimitsPageEmpty2}{\showpaste}
\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch3}
\begin{paste}{ugProblemLimitsPageFull3}{ugProblemLimitsPageEmpty3}
\pastebutton{ugProblemLimitsPageFull3}{\hidepaste}
\tab{5}\spadcommand{limit(x * log(x),x = 0)}
\indentrel{3}\begin{verbatim}
   (3)  [leftHandLimit= "failed",rightHandLimit= 0]
Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty3}
\begin{paste}{ugProblemLimitsPageEmpty3}{ugProblemLimitsPagePatch3}
\pastebutton{ugProblemLimitsPageEmpty3}{\showpaste}
\tab{5}\spadcommand{limit(x * log(x),x = 0)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch4}
\begin{paste}{ugProblemLimitsPageFull4}{ugProblemLimitsPageEmpty4}
\pastebutton{ugProblemLimitsPageFull4}{\hidepaste}
\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
\indentrel{3}\begin{verbatim}
   (4)  [leftHandLimit= - 1,rightHandLimit= 1]
Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty4}
\begin{paste}{ugProblemLimitsPageEmpty4}{ugProblemLimitsPagePatch4}
\pastebutton{ugProblemLimitsPageEmpty4}{\showpaste}
\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch5}
\begin{paste}{ugProblemLimitsPageFull5}{ugProblemLimitsPageEmpty5}
\pastebutton{ugProblemLimitsPageFull5}{\hidepaste}
\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)}
\indentrel{3}\begin{verbatim}
                            1                    1
   (5)  [leftHandLimit= - ÄÄÄÄ,rightHandLimit= ÄÄÄÄ]
                           ÚÄ¿                  ÚÄ¿
                          \³2                  \³2
Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty5}
\begin{paste}{ugProblemLimitsPageEmpty5}{ugProblemLimitsPagePatch5}
\pastebutton{ugProblemLimitsPageEmpty5}{\showpaste}
\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch6}
\begin{paste}{ugProblemLimitsPageFull6}{ugProblemLimitsPageEmpty6}
\pastebutton{ugProblemLimitsPageFull6}{\hidepaste}
\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
\indentrel{3}\begin{verbatim}
         ÚÄ¿
        \³3
   (6)  ÄÄÄÄ
          5
  Type: Union(OrderedCompletion Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty6}
\begin{paste}{ugProblemLimitsPageEmpty6}{ugProblemLimitsPagePatch6}
\pastebutton{ugProblemLimitsPageEmpty6}{\showpaste}
\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch7}
\begin{paste}{ugProblemLimitsPageFull7}{ugProblemLimitsPageEmpty7}
\pastebutton{ugProblemLimitsPageFull7}{\hidepaste}
\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
\indentrel{3}\begin{verbatim}
           ÚÄ¿
          \³3
   (7)  - ÄÄÄÄ
            5
  Type: Union(OrderedCompletion Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty7}
\begin{paste}{ugProblemLimitsPageEmpty7}{ugProblemLimitsPagePatch7}
\pastebutton{ugProblemLimitsPageEmpty7}{\showpaste}
\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch8}
\begin{paste}{ugProblemLimitsPageFull8}{ugProblemLimitsPageEmpty8}
\pastebutton{ugProblemLimitsPageFull8}{\hidepaste}
\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)}
\indentrel{3}\begin{verbatim}
        a
   (8)  Ä
        b
  Type: Union(OrderedCompletion Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty8}
\begin{paste}{ugProblemLimitsPageEmpty8}{ugProblemLimitsPagePatch8}
\pastebutton{ugProblemLimitsPageEmpty8}{\showpaste}
\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch9}
\begin{paste}{ugProblemLimitsPageFull9}{ugProblemLimitsPageEmpty9}
\pastebutton{ugProblemLimitsPageFull9}{\hidepaste}
\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)}
\indentrel{3}\begin{verbatim}
   (9)  0
  Type: Union(OrderedCompletion Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty9}
\begin{paste}{ugProblemLimitsPageEmpty9}{ugProblemLimitsPagePatch9}
\pastebutton{ugProblemLimitsPageEmpty9}{\showpaste}
\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch10}
\begin{paste}{ugProblemLimitsPageFull10}{ugProblemLimitsPageEmpty10}
\pastebutton{ugProblemLimitsPageFull10}{\hidepaste}
\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)}
\indentrel{3}\begin{verbatim}
   (10)  "failed"
                              Type: Union("failed",...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty10}
\begin{paste}{ugProblemLimitsPageEmpty10}{ugProblemLimitsPagePatch10}
\pastebutton{ugProblemLimitsPageEmpty10}{\showpaste}
\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch11}
\begin{paste}{ugProblemLimitsPageFull11}{ugProblemLimitsPageEmpty11}
\pastebutton{ugProblemLimitsPageFull11}{\hidepaste}
\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)}
\indentrel{3}\begin{verbatim}
   (11)  - 1
   Type: OnePointCompletion Fraction Polynomial Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty11}
\begin{paste}{ugProblemLimitsPageEmpty11}{ugProblemLimitsPagePatch11}
\pastebutton{ugProblemLimitsPageEmpty11}{\showpaste}
\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch12}
\begin{paste}{ugProblemLimitsPageFull12}{ugProblemLimitsPageEmpty12}
\pastebutton{ugProblemLimitsPageFull12}{\hidepaste}
\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)}
\indentrel{3}\begin{verbatim}
   (12)  0
  Type: Union(OrderedCompletion Expression Integer,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty12}
\begin{paste}{ugProblemLimitsPageEmpty12}{ugProblemLimitsPagePatch12}
\pastebutton{ugProblemLimitsPageEmpty12}{\showpaste}
\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)}
\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPagePatch13}
\begin{paste}{ugProblemLimitsPageFull13}{ugProblemLimitsPageEmpty13}
\pastebutton{ugProblemLimitsPageFull13}{\hidepaste}
\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)}
\indentrel{3}\begin{verbatim}
   (13)  "failed"
                              Type: Union("failed",...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugProblemLimitsPageEmpty13}
\begin{paste}{ugProblemLimitsPageEmpty13}{ugProblemLimitsPagePatch13}
\pastebutton{ugProblemLimitsPageEmpty13}{\showpaste}
\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)}
\end{paste}\end{patch}