\begin{patch}{RealClosureXmpPagePatch1} \begin{paste}{RealClosureXmpPageFull1}{RealClosureXmpPageEmpty1} \pastebutton{RealClosureXmpPageFull1}{\hidepaste} \tab{5}\spadcommand{Ran := RECLOS(FRAC INT)\bound{Ran }} \indentrel{3}\begin{verbatim} (1) RealClosure Fraction Integer Type: Domain \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty1} \begin{paste}{RealClosureXmpPageEmpty1}{RealClosureXmpPagePatch1} \pastebutton{RealClosureXmpPageEmpty1}{\showpaste} \tab{5}\spadcommand{Ran := RECLOS(FRAC INT)\bound{Ran }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch2} \begin{paste}{RealClosureXmpPageFull2}{RealClosureXmpPageEmpty2} \pastebutton{RealClosureXmpPageFull2}{\hidepaste} \tab{5}\spadcommand{fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran == sqrt(a)+sqrt(b) - sqrt(c)-sqrt(d)\free{Ran }\bound{fs }} \indentrel{3}\begin{verbatim} Type: Void \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty2} \begin{paste}{RealClosureXmpPageEmpty2}{RealClosureXmpPagePatch2} \pastebutton{RealClosureXmpPageEmpty2}{\showpaste} \tab{5}\spadcommand{fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran == sqrt(a)+sqrt(b) - sqrt(c)-sqrt(d)\free{Ran }\bound{fs }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch3} \begin{paste}{RealClosureXmpPageFull3}{RealClosureXmpPageEmpty3} \pastebutton{RealClosureXmpPageFull3}{\hidepaste} \tab{5}\spadcommand{squareDiff1 := fourSquares(73,548,60,586)\free{fs }\bound{sd1 }} \indentrel{3}\begin{verbatim} ���Ŀ ��Ŀ ���Ŀ ��Ŀ (3) - \�586 - \�60 + \�548 + \�73 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty3} \begin{paste}{RealClosureXmpPageEmpty3}{RealClosureXmpPagePatch3} \pastebutton{RealClosureXmpPageEmpty3}{\showpaste} \tab{5}\spadcommand{squareDiff1 := fourSquares(73,548,60,586)\free{fs }\bound{sd1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch4} \begin{paste}{RealClosureXmpPageFull4}{RealClosureXmpPageEmpty4} \pastebutton{RealClosureXmpPageFull4}{\hidepaste} \tab{5}\spadcommand{recip(squareDiff1)\free{sd1 }} \indentrel{3}\begin{verbatim} (4) ���Ŀ ��Ŀ ��Ŀ (54602\�548 + 149602\�73 )\�60 + ��Ŀ ���Ŀ 49502\�73 \�548 + 9900895 * ���Ŀ \�586 + ��Ŀ ���Ŀ ��Ŀ ���Ŀ (154702\�73 \�548 + 30941947)\�60 + 10238421\�548 + ��Ŀ 28051871\�73 Type: Union(RealClosure Fraction Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty4} \begin{paste}{RealClosureXmpPageEmpty4}{RealClosureXmpPagePatch4} \pastebutton{RealClosureXmpPageEmpty4}{\showpaste} \tab{5}\spadcommand{recip(squareDiff1)\free{sd1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch5} \begin{paste}{RealClosureXmpPageFull5}{RealClosureXmpPageEmpty5} \pastebutton{RealClosureXmpPageFull5}{\hidepaste} \tab{5}\spadcommand{sign(squareDiff1)\free{sd1 }} \indentrel{3}\begin{verbatim} (5) 1 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty5} \begin{paste}{RealClosureXmpPageEmpty5}{RealClosureXmpPagePatch5} \pastebutton{RealClosureXmpPageEmpty5}{\showpaste} \tab{5}\spadcommand{sign(squareDiff1)\free{sd1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch6} \begin{paste}{RealClosureXmpPageFull6}{RealClosureXmpPageEmpty6} \pastebutton{RealClosureXmpPageFull6}{\hidepaste} \tab{5}\spadcommand{squareDiff2 := fourSquares(165,778,86,990)\free{fs }\bound{sd2 }} \indentrel{3}\begin{verbatim} ���Ŀ ��Ŀ ���Ŀ ���Ŀ (6) - \�990 - \�86 + \�778 + \�165 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty6} \begin{paste}{RealClosureXmpPageEmpty6}{RealClosureXmpPagePatch6} \pastebutton{RealClosureXmpPageEmpty6}{\showpaste} \tab{5}\spadcommand{squareDiff2 := fourSquares(165,778,86,990)\free{fs }\bound{sd2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch7} \begin{paste}{RealClosureXmpPageFull7}{RealClosureXmpPageEmpty7} \pastebutton{RealClosureXmpPageFull7}{\hidepaste} \tab{5}\spadcommand{recip(squareDiff2)\free{sd2 }} \indentrel{3}\begin{verbatim} (7) ���Ŀ ���Ŀ ��Ŀ (556778\�778 + 1209010\�165 )\�86 + ���Ŀ ���Ŀ 401966\�165 \�778 + 144019431 * ���Ŀ \�990 + ���Ŀ ���Ŀ ��Ŀ (1363822\�165 \�778 + 488640503)\�86 + ���Ŀ ���Ŀ 162460913\�778 + 352774119\�165 Type: Union(RealClosure Fraction Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty7} \begin{paste}{RealClosureXmpPageEmpty7}{RealClosureXmpPagePatch7} \pastebutton{RealClosureXmpPageEmpty7}{\showpaste} \tab{5}\spadcommand{recip(squareDiff2)\free{sd2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch8} \begin{paste}{RealClosureXmpPageFull8}{RealClosureXmpPageEmpty8} \pastebutton{RealClosureXmpPageFull8}{\hidepaste} \tab{5}\spadcommand{sign(squareDiff2)\free{sd2 }} \indentrel{3}\begin{verbatim} (8) 1 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty8} \begin{paste}{RealClosureXmpPageEmpty8}{RealClosureXmpPagePatch8} \pastebutton{RealClosureXmpPageEmpty8}{\showpaste} \tab{5}\spadcommand{sign(squareDiff2)\free{sd2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch9} \begin{paste}{RealClosureXmpPageFull9}{RealClosureXmpPageEmpty9} \pastebutton{RealClosureXmpPageFull9}{\hidepaste} \tab{5}\spadcommand{squareDiff3 := fourSquares(217,708,226,692)\free{fs }\bound{sd3 }} \indentrel{3}\begin{verbatim} ���Ŀ ���Ŀ ���Ŀ ���Ŀ (9) - \�692 - \�226 + \�708 + \�217 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty9} \begin{paste}{RealClosureXmpPageEmpty9}{RealClosureXmpPagePatch9} \pastebutton{RealClosureXmpPageEmpty9}{\showpaste} \tab{5}\spadcommand{squareDiff3 := fourSquares(217,708,226,692)\free{fs }\bound{sd3 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch10} \begin{paste}{RealClosureXmpPageFull10}{RealClosureXmpPageEmpty10} \pastebutton{RealClosureXmpPageFull10}{\hidepaste} \tab{5}\spadcommand{recip(squareDiff3)\free{sd3 }} \indentrel{3}\begin{verbatim} (10) ���Ŀ ���Ŀ ���Ŀ (- 34102\�708 - 61598\�217 )\�226 + ���Ŀ ���Ŀ - 34802\�217 \�708 - 13641141 * ���Ŀ \�692 + ���Ŀ ���Ŀ ���Ŀ (- 60898\�217 \�708 - 23869841)\�226 + ���Ŀ ���Ŀ - 13486123\�708 - 24359809\�217 Type: Union(RealClosure Fraction Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty10} \begin{paste}{RealClosureXmpPageEmpty10}{RealClosureXmpPagePatch10} \pastebutton{RealClosureXmpPageEmpty10}{\showpaste} \tab{5}\spadcommand{recip(squareDiff3)\free{sd3 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch11} \begin{paste}{RealClosureXmpPageFull11}{RealClosureXmpPageEmpty11} \pastebutton{RealClosureXmpPageFull11}{\hidepaste} \tab{5}\spadcommand{sign(squareDiff3)\free{sd3 }} \indentrel{3}\begin{verbatim} (11) - 1 Type: Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty11} \begin{paste}{RealClosureXmpPageEmpty11}{RealClosureXmpPagePatch11} \pastebutton{RealClosureXmpPageEmpty11}{\showpaste} \tab{5}\spadcommand{sign(squareDiff3)\free{sd3 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch12} \begin{paste}{RealClosureXmpPageFull12}{RealClosureXmpPageEmpty12} \pastebutton{RealClosureXmpPageFull12}{\hidepaste} \tab{5}\spadcommand{squareDiff4 := fourSquares(155,836,162,820)\free{fs }\bound{sd4 }} \indentrel{3}\begin{verbatim} ���Ŀ ���Ŀ ���Ŀ ���Ŀ (12) - \�820 - \�162 + \�836 + \�155 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty12} \begin{paste}{RealClosureXmpPageEmpty12}{RealClosureXmpPagePatch12} \pastebutton{RealClosureXmpPageEmpty12}{\showpaste} \tab{5}\spadcommand{squareDiff4 := fourSquares(155,836,162,820)\free{fs }\bound{sd4 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch13} \begin{paste}{RealClosureXmpPageFull13}{RealClosureXmpPageEmpty13} \pastebutton{RealClosureXmpPageFull13}{\hidepaste} \tab{5}\spadcommand{recip(squareDiff4)\free{sd4 }} \indentrel{3}\begin{verbatim} (13) ���Ŀ ���Ŀ ���Ŀ (- 37078\�836 - 86110\�155 )\�162 + ���Ŀ ���Ŀ - 37906\�155 \�836 - 13645107 * ���Ŀ \�820 + ���Ŀ ���Ŀ ���Ŀ (- 85282\�155 \�836 - 30699151)\�162 + ���Ŀ ���Ŀ - 13513901\�836 - 31384703\�155 Type: Union(RealClosure Fraction Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty13} \begin{paste}{RealClosureXmpPageEmpty13}{RealClosureXmpPagePatch13} \pastebutton{RealClosureXmpPageEmpty13}{\showpaste} \tab{5}\spadcommand{recip(squareDiff4)\free{sd4 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch14} \begin{paste}{RealClosureXmpPageFull14}{RealClosureXmpPageEmpty14} \pastebutton{RealClosureXmpPageFull14}{\hidepaste} \tab{5}\spadcommand{sign(squareDiff4)\free{sd4 }} \indentrel{3}\begin{verbatim} (14) - 1 Type: Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty14} \begin{paste}{RealClosureXmpPageEmpty14}{RealClosureXmpPagePatch14} \pastebutton{RealClosureXmpPageEmpty14}{\showpaste} \tab{5}\spadcommand{sign(squareDiff4)\free{sd4 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch15} \begin{paste}{RealClosureXmpPageFull15}{RealClosureXmpPageEmpty15} \pastebutton{RealClosureXmpPageFull15}{\hidepaste} \tab{5}\spadcommand{squareDiff5 := fourSquares(591,772,552,818)\free{fs }\bound{sd5 }} \indentrel{3}\begin{verbatim} ���Ŀ ���Ŀ ���Ŀ ���Ŀ (15) - \�818 - \�552 + \�772 + \�591 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty15} \begin{paste}{RealClosureXmpPageEmpty15}{RealClosureXmpPagePatch15} \pastebutton{RealClosureXmpPageEmpty15}{\showpaste} \tab{5}\spadcommand{squareDiff5 := fourSquares(591,772,552,818)\free{fs }\bound{sd5 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch16} \begin{paste}{RealClosureXmpPageFull16}{RealClosureXmpPageEmpty16} \pastebutton{RealClosureXmpPageFull16}{\hidepaste} \tab{5}\spadcommand{recip(squareDiff5)\free{sd5 }} \indentrel{3}\begin{verbatim} (16) ���Ŀ ���Ŀ ���Ŀ (70922\�772 + 81058\�591 )\�552 + ���Ŀ ���Ŀ 68542\�591 \�772 + 46297673 * ���Ŀ \�818 + ���Ŀ ���Ŀ ���Ŀ ���Ŀ (83438\�591 \�772 + 56359389)\�552 + 47657051\�772 + ���Ŀ 54468081\�591 Type: Union(RealClosure Fraction Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty16} \begin{paste}{RealClosureXmpPageEmpty16}{RealClosureXmpPagePatch16} \pastebutton{RealClosureXmpPageEmpty16}{\showpaste} \tab{5}\spadcommand{recip(squareDiff5)\free{sd5 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch17} \begin{paste}{RealClosureXmpPageFull17}{RealClosureXmpPageEmpty17} \pastebutton{RealClosureXmpPageFull17}{\hidepaste} \tab{5}\spadcommand{sign(squareDiff5)\free{sd5 }} \indentrel{3}\begin{verbatim} (17) 1 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty17} \begin{paste}{RealClosureXmpPageEmpty17}{RealClosureXmpPagePatch17} \pastebutton{RealClosureXmpPageEmpty17}{\showpaste} \tab{5}\spadcommand{sign(squareDiff5)\free{sd5 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch18} \begin{paste}{RealClosureXmpPageFull18}{RealClosureXmpPageEmpty18} \pastebutton{RealClosureXmpPageFull18}{\hidepaste} \tab{5}\spadcommand{squareDiff6 := fourSquares(434,1053,412,1088)\free{fs }\bound{sd6 }} \indentrel{3}\begin{verbatim} ����Ŀ ���Ŀ ����Ŀ ���Ŀ (18) - \�1088 - \�412 + \�1053 + \�434 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty18} \begin{paste}{RealClosureXmpPageEmpty18}{RealClosureXmpPagePatch18} \pastebutton{RealClosureXmpPageEmpty18}{\showpaste} \tab{5}\spadcommand{squareDiff6 := fourSquares(434,1053,412,1088)\free{fs }\bound{sd6 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch19} \begin{paste}{RealClosureXmpPageFull19}{RealClosureXmpPageEmpty19} \pastebutton{RealClosureXmpPageFull19}{\hidepaste} \tab{5}\spadcommand{recip(squareDiff6)\free{sd6 }} \indentrel{3}\begin{verbatim} (19) ����Ŀ ���Ŀ ���Ŀ (115442\�1053 + 179818\�434 )\�412 + ���Ŀ ����Ŀ 112478\�434 \�1053 + 76037291 * ����Ŀ \�1088 + ���Ŀ ����Ŀ ���Ŀ (182782\�434 \�1053 + 123564147)\�412 + ����Ŀ ���Ŀ 77290639\�1053 + 120391609\�434 Type: Union(RealClosure Fraction Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty19} \begin{paste}{RealClosureXmpPageEmpty19}{RealClosureXmpPagePatch19} \pastebutton{RealClosureXmpPageEmpty19}{\showpaste} \tab{5}\spadcommand{recip(squareDiff6)\free{sd6 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch20} \begin{paste}{RealClosureXmpPageFull20}{RealClosureXmpPageEmpty20} \pastebutton{RealClosureXmpPageFull20}{\hidepaste} \tab{5}\spadcommand{sign(squareDiff6)\free{sd6 }} \indentrel{3}\begin{verbatim} (20) 1 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty20} \begin{paste}{RealClosureXmpPageEmpty20}{RealClosureXmpPagePatch20} \pastebutton{RealClosureXmpPageEmpty20}{\showpaste} \tab{5}\spadcommand{sign(squareDiff6)\free{sd6 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch21} \begin{paste}{RealClosureXmpPageFull21}{RealClosureXmpPageEmpty21} \pastebutton{RealClosureXmpPageFull21}{\hidepaste} \tab{5}\spadcommand{squareDiff7 := fourSquares(514,1049,446,1152)\free{fs }\bound{sd7 }} \indentrel{3}\begin{verbatim} ����Ŀ ���Ŀ ����Ŀ ���Ŀ (21) - \�1152 - \�446 + \�1049 + \�514 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty21} \begin{paste}{RealClosureXmpPageEmpty21}{RealClosureXmpPagePatch21} \pastebutton{RealClosureXmpPageEmpty21}{\showpaste} \tab{5}\spadcommand{squareDiff7 := fourSquares(514,1049,446,1152)\free{fs }\bound{sd7 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch22} \begin{paste}{RealClosureXmpPageFull22}{RealClosureXmpPageEmpty22} \pastebutton{RealClosureXmpPageFull22}{\hidepaste} \tab{5}\spadcommand{recip(squareDiff7)\free{sd7 }} \indentrel{3}\begin{verbatim} (22) ����Ŀ ���Ŀ ���Ŀ (349522\�1049 + 499322\�514 )\�446 + ���Ŀ ����Ŀ 325582\�514 \�1049 + 239072537 * ����Ŀ \�1152 + ���Ŀ ����Ŀ ���Ŀ (523262\�514 \�1049 + 384227549)\�446 + ����Ŀ ���Ŀ 250534873\�1049 + 357910443\�514 Type: Union(RealClosure Fraction Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty22} \begin{paste}{RealClosureXmpPageEmpty22}{RealClosureXmpPagePatch22} \pastebutton{RealClosureXmpPageEmpty22}{\showpaste} \tab{5}\spadcommand{recip(squareDiff7)\free{sd7 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch23} \begin{paste}{RealClosureXmpPageFull23}{RealClosureXmpPageEmpty23} \pastebutton{RealClosureXmpPageFull23}{\hidepaste} \tab{5}\spadcommand{sign(squareDiff7)\free{sd7 }} \indentrel{3}\begin{verbatim} (23) 1 Type: PositiveInteger \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty23} \begin{paste}{RealClosureXmpPageEmpty23}{RealClosureXmpPagePatch23} \pastebutton{RealClosureXmpPageEmpty23}{\showpaste} \tab{5}\spadcommand{sign(squareDiff7)\free{sd7 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch24} \begin{paste}{RealClosureXmpPageFull24}{RealClosureXmpPageEmpty24} \pastebutton{RealClosureXmpPageFull24}{\hidepaste} \tab{5}\spadcommand{squareDiff8 := fourSquares(190,1751,208,1698)\free{fs }\bound{sd8 }} \indentrel{3}\begin{verbatim} ����Ŀ ���Ŀ ����Ŀ ���Ŀ (24) - \�1698 - \�208 + \�1751 + \�190 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty24} \begin{paste}{RealClosureXmpPageEmpty24}{RealClosureXmpPagePatch24} \pastebutton{RealClosureXmpPageEmpty24}{\showpaste} \tab{5}\spadcommand{squareDiff8 := fourSquares(190,1751,208,1698)\free{fs }\bound{sd8 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch25} \begin{paste}{RealClosureXmpPageFull25}{RealClosureXmpPageEmpty25} \pastebutton{RealClosureXmpPageFull25}{\hidepaste} \tab{5}\spadcommand{recip(squareDiff8)\free{sd8 }} \indentrel{3}\begin{verbatim} (25) ����Ŀ ���Ŀ ���Ŀ (- 214702\�1751 - 651782\�190 )\�208 + ���Ŀ ����Ŀ - 224642\�190 \�1751 - 129571901 * ����Ŀ \�1698 + ���Ŀ ����Ŀ ���Ŀ (- 641842\�190 \�1751 - 370209881)\�208 + ����Ŀ ���Ŀ - 127595865\�1751 - 387349387\�190 Type: Union(RealClosure Fraction Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty25} \begin{paste}{RealClosureXmpPageEmpty25}{RealClosureXmpPagePatch25} \pastebutton{RealClosureXmpPageEmpty25}{\showpaste} \tab{5}\spadcommand{recip(squareDiff8)\free{sd8 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch26} \begin{paste}{RealClosureXmpPageFull26}{RealClosureXmpPageEmpty26} \pastebutton{RealClosureXmpPageFull26}{\hidepaste} \tab{5}\spadcommand{sign(squareDiff8)\free{sd8 }} \indentrel{3}\begin{verbatim} (26) - 1 Type: Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty26} \begin{paste}{RealClosureXmpPageEmpty26}{RealClosureXmpPagePatch26} \pastebutton{RealClosureXmpPageEmpty26}{\showpaste} \tab{5}\spadcommand{sign(squareDiff8)\free{sd8 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch27} \begin{paste}{RealClosureXmpPageFull27}{RealClosureXmpPageEmpty27} \pastebutton{RealClosureXmpPageFull27}{\hidepaste} \tab{5}\spadcommand{relativeApprox(squareDiff8,10**(-3))::Float\free{sd8 }} \indentrel{3}\begin{verbatim} (27) - 0.2340527771 5937700123 E -10 Type: Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty27} \begin{paste}{RealClosureXmpPageEmpty27}{RealClosureXmpPagePatch27} \pastebutton{RealClosureXmpPageEmpty27}{\showpaste} \tab{5}\spadcommand{relativeApprox(squareDiff8,10**(-3))::Float\free{sd8 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch28} \begin{paste}{RealClosureXmpPageFull28}{RealClosureXmpPageEmpty28} \pastebutton{RealClosureXmpPageFull28}{\hidepaste} \tab{5}\spadcommand{l := allRootsOf((x**2-2)**2-2)$Ran\free{Ran }\bound{l }} \indentrel{3}\begin{verbatim} (28) [%R33,%R34,%R35,%R36] Type: List RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty28} \begin{paste}{RealClosureXmpPageEmpty28}{RealClosureXmpPagePatch28} \pastebutton{RealClosureXmpPageEmpty28}{\showpaste} \tab{5}\spadcommand{l := allRootsOf((x**2-2)**2-2)$Ran\free{Ran }\bound{l }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch29} \begin{paste}{RealClosureXmpPageFull29}{RealClosureXmpPageEmpty29} \pastebutton{RealClosureXmpPageFull29}{\hidepaste} \tab{5}\spadcommand{removeDuplicates map(mainDefiningPolynomial,l)\free{l }} \indentrel{3}\begin{verbatim} 4 2 (29) [? - 4? + 2] Type: List Union(SparseUnivariatePolynomial RealClosure Fraction Integer,"failed") \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty29} \begin{paste}{RealClosureXmpPageEmpty29}{RealClosureXmpPagePatch29} \pastebutton{RealClosureXmpPageEmpty29}{\showpaste} \tab{5}\spadcommand{removeDuplicates map(mainDefiningPolynomial,l)\free{l }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch30} \begin{paste}{RealClosureXmpPageFull30}{RealClosureXmpPageEmpty30} \pastebutton{RealClosureXmpPageFull30}{\hidepaste} \tab{5}\spadcommand{map(mainCharacterization,l)\free{l }} \indentrel{3}\begin{verbatim} (30) [[- 2,- 1[,[- 1,0[,[0,1[,[1,2[] Type: List Union(RightOpenIntervalRootCharacterization(RealClosure Fraction Integer,SparseUnivariatePolynomial RealClosure Fraction Integer),"failed") \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty30} \begin{paste}{RealClosureXmpPageEmpty30}{RealClosureXmpPagePatch30} \pastebutton{RealClosureXmpPageEmpty30}{\showpaste} \tab{5}\spadcommand{map(mainCharacterization,l)\free{l }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch31} \begin{paste}{RealClosureXmpPageFull31}{RealClosureXmpPageEmpty31} \pastebutton{RealClosureXmpPageFull31}{\hidepaste} \tab{5}\spadcommand{[reduce(+,l),reduce(*,l)-2]\free{l }} \indentrel{3}\begin{verbatim} (31) [0,0] Type: List RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty31} \begin{paste}{RealClosureXmpPageEmpty31}{RealClosureXmpPagePatch31} \pastebutton{RealClosureXmpPageEmpty31}{\showpaste} \tab{5}\spadcommand{[reduce(+,l),reduce(*,l)-2]\free{l }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch32} \begin{paste}{RealClosureXmpPageFull32}{RealClosureXmpPageEmpty32} \pastebutton{RealClosureXmpPageFull32}{\hidepaste} \tab{5}\spadcommand{(s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran)\free{Ran }\bound{s2 }\bound{s5 }\bound{s10 }} \indentrel{3}\begin{verbatim} ��Ŀ (32) \�10 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty32} \begin{paste}{RealClosureXmpPageEmpty32}{RealClosureXmpPagePatch32} \pastebutton{RealClosureXmpPageEmpty32}{\showpaste} \tab{5}\spadcommand{(s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran)\free{Ran }\bound{s2 }\bound{s5 }\bound{s10 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch33} \begin{paste}{RealClosureXmpPageFull33}{RealClosureXmpPageEmpty33} \pastebutton{RealClosureXmpPageFull33}{\hidepaste} \tab{5}\spadcommand{eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10)\free{s2 }\free{s5 }\free{s10 }\bound{eq1 }} \indentrel{3}\begin{verbatim} (33) ���������Ŀ ��������Ŀ ���������Ŀ ��������Ŀ � ��Ŀ � �Ŀ � ��Ŀ � �Ŀ - \�\�10 - 3 \�\�5 - 2 + \�\�10 + 3 \�\�5 + 2 = �����������Ŀ � �Ŀ \�10\�2 + 10 Type: Equation RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty33} \begin{paste}{RealClosureXmpPageEmpty33}{RealClosureXmpPagePatch33} \pastebutton{RealClosureXmpPageEmpty33}{\showpaste} \tab{5}\spadcommand{eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10)\free{s2 }\free{s5 }\free{s10 }\bound{eq1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch34} \begin{paste}{RealClosureXmpPageFull34}{RealClosureXmpPageEmpty34} \pastebutton{RealClosureXmpPageFull34}{\hidepaste} \tab{5}\spadcommand{eq1::Boolean\free{eq1 }} \indentrel{3}\begin{verbatim} (34) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty34} \begin{paste}{RealClosureXmpPageEmpty34}{RealClosureXmpPagePatch34} \pastebutton{RealClosureXmpPageEmpty34}{\showpaste} \tab{5}\spadcommand{eq1::Boolean\free{eq1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch35} \begin{paste}{RealClosureXmpPageFull35}{RealClosureXmpPageEmpty35} \pastebutton{RealClosureXmpPageFull35}{\hidepaste} \tab{5}\spadcommand{eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)\free{s2 }\free{s5 }\free{s10 }\bound{eq2 }} \indentrel{3}\begin{verbatim} (35) ��������Ŀ ��������Ŀ ��������Ŀ ��������Ŀ � �Ŀ � �Ŀ � �Ŀ � �Ŀ - \�\�5 - 2 \�\�2 - 1 + \�\�5 + 2 \�\�2 + 1 = ����������Ŀ � ��Ŀ \�2\�10 + 2 Type: Equation RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty35} \begin{paste}{RealClosureXmpPageEmpty35}{RealClosureXmpPagePatch35} \pastebutton{RealClosureXmpPageEmpty35}{\showpaste} \tab{5}\spadcommand{eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)\free{s2 }\free{s5 }\free{s10 }\bound{eq2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch36} \begin{paste}{RealClosureXmpPageFull36}{RealClosureXmpPageEmpty36} \pastebutton{RealClosureXmpPageFull36}{\hidepaste} \tab{5}\spadcommand{eq2::Boolean\free{eq2 }} \indentrel{3}\begin{verbatim} (36) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty36} \begin{paste}{RealClosureXmpPageEmpty36}{RealClosureXmpPagePatch36} \pastebutton{RealClosureXmpPageEmpty36}{\showpaste} \tab{5}\spadcommand{eq2::Boolean\free{eq2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch37} \begin{paste}{RealClosureXmpPageFull37}{RealClosureXmpPageEmpty37} \pastebutton{RealClosureXmpPageFull37}{\hidepaste} \tab{5}\spadcommand{s3 := sqrt(3)$Ran\free{Ran }\bound{s3 }} \indentrel{3}\begin{verbatim} �Ŀ (37) \�3 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty37} \begin{paste}{RealClosureXmpPageEmpty37}{RealClosureXmpPagePatch37} \pastebutton{RealClosureXmpPageEmpty37}{\showpaste} \tab{5}\spadcommand{s3 := sqrt(3)$Ran\free{Ran }\bound{s3 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch38} \begin{paste}{RealClosureXmpPageFull38}{RealClosureXmpPageEmpty38} \pastebutton{RealClosureXmpPageFull38}{\hidepaste} \tab{5}\spadcommand{s7:= sqrt(7)$Ran\free{Ran }\bound{s7 }} \indentrel{3}\begin{verbatim} �Ŀ (38) \�7 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty38} \begin{paste}{RealClosureXmpPageEmpty38}{RealClosureXmpPagePatch38} \pastebutton{RealClosureXmpPageEmpty38}{\showpaste} \tab{5}\spadcommand{s7:= sqrt(7)$Ran\free{Ran }\bound{s7 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch39} \begin{paste}{RealClosureXmpPageFull39}{RealClosureXmpPageEmpty39} \pastebutton{RealClosureXmpPageFull39}{\hidepaste} \tab{5}\spadcommand{e1 := sqrt(2*s7-3*s3,3)\free{s7 }\free{s3 }\bound{e1 }} \indentrel{3}\begin{verbatim} �������������Ŀ 3� �Ŀ �Ŀ (39) \�2\�7 - 3\�3 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty39} \begin{paste}{RealClosureXmpPageEmpty39}{RealClosureXmpPagePatch39} \pastebutton{RealClosureXmpPageEmpty39}{\showpaste} \tab{5}\spadcommand{e1 := sqrt(2*s7-3*s3,3)\free{s7 }\free{s3 }\bound{e1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch40} \begin{paste}{RealClosureXmpPageFull40}{RealClosureXmpPageEmpty40} \pastebutton{RealClosureXmpPageFull40}{\hidepaste} \tab{5}\spadcommand{e2 := sqrt(2*s7+3*s3,3)\free{s7 }\free{s3 }\bound{e2 }} \indentrel{3}\begin{verbatim} �������������Ŀ 3� �Ŀ �Ŀ (40) \�2\�7 + 3\�3 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty40} \begin{paste}{RealClosureXmpPageEmpty40}{RealClosureXmpPagePatch40} \pastebutton{RealClosureXmpPageEmpty40}{\showpaste} \tab{5}\spadcommand{e2 := sqrt(2*s7+3*s3,3)\free{s7 }\free{s3 }\bound{e2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch41} \begin{paste}{RealClosureXmpPageFull41}{RealClosureXmpPageEmpty41} \pastebutton{RealClosureXmpPageFull41}{\hidepaste} \tab{5}\spadcommand{e2-e1-s3\free{e2 }\free{e1 }\free{s3 }} \indentrel{3}\begin{verbatim} (41) 0 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty41} \begin{paste}{RealClosureXmpPageEmpty41}{RealClosureXmpPagePatch41} \pastebutton{RealClosureXmpPageEmpty41}{\showpaste} \tab{5}\spadcommand{e2-e1-s3\free{e2 }\free{e1 }\free{s3 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch42} \begin{paste}{RealClosureXmpPageFull42}{RealClosureXmpPageEmpty42} \pastebutton{RealClosureXmpPageFull42}{\hidepaste} \tab{5}\spadcommand{pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3)\free{Ran }\bound{pol }} \indentrel{3}\begin{verbatim} 4 7 2 100 (42) x + � x + 30x - ��� 3 3 Type: UnivariatePolynomial(x,RealClosure Fraction Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty42} \begin{paste}{RealClosureXmpPageEmpty42}{RealClosureXmpPagePatch42} \pastebutton{RealClosureXmpPageEmpty42}{\showpaste} \tab{5}\spadcommand{pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3)\free{Ran }\bound{pol }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch43} \begin{paste}{RealClosureXmpPageFull43}{RealClosureXmpPageEmpty43} \pastebutton{RealClosureXmpPageFull43}{\hidepaste} \tab{5}\spadcommand{r1 := sqrt(7633)$Ran\free{Ran }\bound{r1 }} \indentrel{3}\begin{verbatim} ����Ŀ (43) \�7633 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty43} \begin{paste}{RealClosureXmpPageEmpty43}{RealClosureXmpPagePatch43} \pastebutton{RealClosureXmpPageEmpty43}{\showpaste} \tab{5}\spadcommand{r1 := sqrt(7633)$Ran\free{Ran }\bound{r1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch44} \begin{paste}{RealClosureXmpPageFull44}{RealClosureXmpPageEmpty44} \pastebutton{RealClosureXmpPageFull44}{\hidepaste} \tab{5}\spadcommand{alpha := sqrt(5*r1-436,3)/3\free{r1 }\bound{alpha }} \indentrel{3}\begin{verbatim} ��������������Ŀ 1 3� ����Ŀ (44) � \�5\�7633 - 436 3 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty44} \begin{paste}{RealClosureXmpPageEmpty44}{RealClosureXmpPagePatch44} \pastebutton{RealClosureXmpPageEmpty44}{\showpaste} \tab{5}\spadcommand{alpha := sqrt(5*r1-436,3)/3\free{r1 }\bound{alpha }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch45} \begin{paste}{RealClosureXmpPageFull45}{RealClosureXmpPageEmpty45} \pastebutton{RealClosureXmpPageFull45}{\hidepaste} \tab{5}\spadcommand{beta := -sqrt(5*r1+436,3)/3\free{r1 }\bound{beta }} \indentrel{3}\begin{verbatim} ��������������Ŀ 1 3� ����Ŀ (45) - � \�5\�7633 + 436 3 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty45} \begin{paste}{RealClosureXmpPageEmpty45}{RealClosureXmpPagePatch45} \pastebutton{RealClosureXmpPageEmpty45}{\showpaste} \tab{5}\spadcommand{beta := -sqrt(5*r1+436,3)/3\free{r1 }\bound{beta }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch46} \begin{paste}{RealClosureXmpPageFull46}{RealClosureXmpPageEmpty46} \pastebutton{RealClosureXmpPageFull46}{\hidepaste} \tab{5}\spadcommand{pol.(alpha+beta-1/3)\free{pol }\free{alpha }\free{beta }} \indentrel{3}\begin{verbatim} (46) 0 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty46} \begin{paste}{RealClosureXmpPageEmpty46}{RealClosureXmpPagePatch46} \pastebutton{RealClosureXmpPageEmpty46}{\showpaste} \tab{5}\spadcommand{pol.(alpha+beta-1/3)\free{pol }\free{alpha }\free{beta }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch47} \begin{paste}{RealClosureXmpPageFull47}{RealClosureXmpPageEmpty47} \pastebutton{RealClosureXmpPageFull47}{\hidepaste} \tab{5}\spadcommand{qol : UP(x,Ran) := x**5+10*x**3+20*x+22\free{Ran }\bound{qol }} \indentrel{3}\begin{verbatim} 5 3 (47) x + 10x + 20x + 22 Type: UnivariatePolynomial(x,RealClosure Fraction Integer) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty47} \begin{paste}{RealClosureXmpPageEmpty47}{RealClosureXmpPagePatch47} \pastebutton{RealClosureXmpPageEmpty47}{\showpaste} \tab{5}\spadcommand{qol : UP(x,Ran) := x**5+10*x**3+20*x+22\free{Ran }\bound{qol }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch48} \begin{paste}{RealClosureXmpPageFull48}{RealClosureXmpPageEmpty48} \pastebutton{RealClosureXmpPageFull48}{\hidepaste} \tab{5}\spadcommand{r2 := sqrt(153)$Ran\free{Ran }\bound{r2 }} \indentrel{3}\begin{verbatim} ���Ŀ (48) \�153 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty48} \begin{paste}{RealClosureXmpPageEmpty48}{RealClosureXmpPagePatch48} \pastebutton{RealClosureXmpPageEmpty48}{\showpaste} \tab{5}\spadcommand{r2 := sqrt(153)$Ran\free{Ran }\bound{r2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch49} \begin{paste}{RealClosureXmpPageFull49}{RealClosureXmpPageEmpty49} \pastebutton{RealClosureXmpPageFull49}{\hidepaste} \tab{5}\spadcommand{alpha2 := sqrt(r2-11,5)\free{r2 }\bound{alpha2 }} \indentrel{3}\begin{verbatim} �����������Ŀ 5� ���Ŀ (49) \�\�153 - 11 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty49} \begin{paste}{RealClosureXmpPageEmpty49}{RealClosureXmpPagePatch49} \pastebutton{RealClosureXmpPageEmpty49}{\showpaste} \tab{5}\spadcommand{alpha2 := sqrt(r2-11,5)\free{r2 }\bound{alpha2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch50} \begin{paste}{RealClosureXmpPageFull50}{RealClosureXmpPageEmpty50} \pastebutton{RealClosureXmpPageFull50}{\hidepaste} \tab{5}\spadcommand{beta2 := -sqrt(r2+11,5)\free{r2 }\bound{beta2 }} \indentrel{3}\begin{verbatim} �����������Ŀ 5� ���Ŀ (50) - \�\�153 + 11 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty50} \begin{paste}{RealClosureXmpPageEmpty50}{RealClosureXmpPagePatch50} \pastebutton{RealClosureXmpPageEmpty50}{\showpaste} \tab{5}\spadcommand{beta2 := -sqrt(r2+11,5)\free{r2 }\bound{beta2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch51} \begin{paste}{RealClosureXmpPageFull51}{RealClosureXmpPageEmpty51} \pastebutton{RealClosureXmpPageFull51}{\hidepaste} \tab{5}\spadcommand{qol(alpha2+beta2)\free{qol }\free{alpha2 }\free{beta2 }} \indentrel{3}\begin{verbatim} (51) 0 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty51} \begin{paste}{RealClosureXmpPageEmpty51}{RealClosureXmpPagePatch51} \pastebutton{RealClosureXmpPageEmpty51}{\showpaste} \tab{5}\spadcommand{qol(alpha2+beta2)\free{qol }\free{alpha2 }\free{beta2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch52} \begin{paste}{RealClosureXmpPageFull52}{RealClosureXmpPageEmpty52} \pastebutton{RealClosureXmpPageFull52}{\hidepaste} \tab{5}\spadcommand{dst1:=sqrt(9+4*s2)=1+2*s2\free{s2 }\bound{dst1 }} \indentrel{3}\begin{verbatim} ���������Ŀ � �Ŀ �Ŀ (52) \�4\�2 + 9 = 2\�2 + 1 Type: Equation RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty52} \begin{paste}{RealClosureXmpPageEmpty52}{RealClosureXmpPagePatch52} \pastebutton{RealClosureXmpPageEmpty52}{\showpaste} \tab{5}\spadcommand{dst1:=sqrt(9+4*s2)=1+2*s2\free{s2 }\bound{dst1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch53} \begin{paste}{RealClosureXmpPageFull53}{RealClosureXmpPageEmpty53} \pastebutton{RealClosureXmpPageFull53}{\hidepaste} \tab{5}\spadcommand{dst1::Boolean\free{dst1 }} \indentrel{3}\begin{verbatim} (53) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty53} \begin{paste}{RealClosureXmpPageEmpty53}{RealClosureXmpPagePatch53} \pastebutton{RealClosureXmpPageEmpty53}{\showpaste} \tab{5}\spadcommand{dst1::Boolean\free{dst1 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch54} \begin{paste}{RealClosureXmpPageFull54}{RealClosureXmpPageEmpty54} \pastebutton{RealClosureXmpPageFull54}{\hidepaste} \tab{5}\spadcommand{s6:Ran:=sqrt 6\free{Ran }\bound{s6 }} \indentrel{3}\begin{verbatim} �Ŀ (54) \�6 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty54} \begin{paste}{RealClosureXmpPageEmpty54}{RealClosureXmpPagePatch54} \pastebutton{RealClosureXmpPageEmpty54}{\showpaste} \tab{5}\spadcommand{s6:Ran:=sqrt 6\free{Ran }\bound{s6 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch55} \begin{paste}{RealClosureXmpPageFull55}{RealClosureXmpPageEmpty55} \pastebutton{RealClosureXmpPageFull55}{\hidepaste} \tab{5}\spadcommand{dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3\free{s6 }\free{s3 }\bound{dst2 }} \indentrel{3}\begin{verbatim} �����������Ŀ ���������Ŀ � �Ŀ � �Ŀ �Ŀ (55) \�- 2\�6 + 5 + \�2\�6 + 5 = 2\�3 Type: Equation RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty55} \begin{paste}{RealClosureXmpPageEmpty55}{RealClosureXmpPagePatch55} \pastebutton{RealClosureXmpPageEmpty55}{\showpaste} \tab{5}\spadcommand{dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3\free{s6 }\free{s3 }\bound{dst2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch56} \begin{paste}{RealClosureXmpPageFull56}{RealClosureXmpPageEmpty56} \pastebutton{RealClosureXmpPageFull56}{\hidepaste} \tab{5}\spadcommand{dst2::Boolean\free{dst2 }} \indentrel{3}\begin{verbatim} (56) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty56} \begin{paste}{RealClosureXmpPageEmpty56}{RealClosureXmpPagePatch56} \pastebutton{RealClosureXmpPageEmpty56}{\showpaste} \tab{5}\spadcommand{dst2::Boolean\free{dst2 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch57} \begin{paste}{RealClosureXmpPageFull57}{RealClosureXmpPageEmpty57} \pastebutton{RealClosureXmpPageFull57}{\hidepaste} \tab{5}\spadcommand{s29:Ran:=sqrt 29\free{Ran }\bound{s29 }} \indentrel{3}\begin{verbatim} ��Ŀ (57) \�29 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty57} \begin{paste}{RealClosureXmpPageEmpty57}{RealClosureXmpPagePatch57} \pastebutton{RealClosureXmpPageEmpty57}{\showpaste} \tab{5}\spadcommand{s29:Ran:=sqrt 29\free{Ran }\bound{s29 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch58} \begin{paste}{RealClosureXmpPageFull58}{RealClosureXmpPageEmpty58} \pastebutton{RealClosureXmpPageFull58}{\hidepaste} \tab{5}\spadcommand{dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5\free{s29 }\free{s5 }\bound{dst4 }} \indentrel{3}\begin{verbatim} (58) ��������������������������������Ŀ � ��������������Ŀ � � ��Ŀ ��Ŀ \�2\�- 10\�29 + 55 - 2\�29 + 16 = �����������Ŀ ����������Ŀ � ��Ŀ � �Ŀ �Ŀ - \�2\�29 + 11 + \�2\�5 + 22 + \�5 Type: Equation RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty58} \begin{paste}{RealClosureXmpPageEmpty58}{RealClosureXmpPagePatch58} \pastebutton{RealClosureXmpPageEmpty58}{\showpaste} \tab{5}\spadcommand{dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5\free{s29 }\free{s5 }\bound{dst4 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch59} \begin{paste}{RealClosureXmpPageFull59}{RealClosureXmpPageEmpty59} \pastebutton{RealClosureXmpPageFull59}{\hidepaste} \tab{5}\spadcommand{dst4::Boolean\free{dst4 }} \indentrel{3}\begin{verbatim} (59) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty59} \begin{paste}{RealClosureXmpPageEmpty59}{RealClosureXmpPagePatch59} \pastebutton{RealClosureXmpPageEmpty59}{\showpaste} \tab{5}\spadcommand{dst4::Boolean\free{dst4 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch60} \begin{paste}{RealClosureXmpPageFull60}{RealClosureXmpPageEmpty60} \pastebutton{RealClosureXmpPageFull60}{\hidepaste} \tab{5}\spadcommand{dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5\free{s2 }\free{s5 }\bound{dst6 }} \indentrel{3}\begin{verbatim} (60) ��������������������������������Ŀ � �Ŀ �Ŀ �Ŀ \�(34\�2 + 46)\�5 + 70\�2 + 112 = �Ŀ �Ŀ �Ŀ (\�2 + 3)\�5 + 4\�2 + 5 Type: Equation RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty60} \begin{paste}{RealClosureXmpPageEmpty60}{RealClosureXmpPagePatch60} \pastebutton{RealClosureXmpPageEmpty60}{\showpaste} \tab{5}\spadcommand{dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5\free{s2 }\free{s5 }\bound{dst6 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch61} \begin{paste}{RealClosureXmpPageFull61}{RealClosureXmpPageEmpty61} \pastebutton{RealClosureXmpPageFull61}{\hidepaste} \tab{5}\spadcommand{dst6::Boolean\free{dst6 }} \indentrel{3}\begin{verbatim} (61) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty61} \begin{paste}{RealClosureXmpPageEmpty61}{RealClosureXmpPagePatch61} \pastebutton{RealClosureXmpPageEmpty61}{\showpaste} \tab{5}\spadcommand{dst6::Boolean\free{dst6 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch62} \begin{paste}{RealClosureXmpPageFull62}{RealClosureXmpPageEmpty62} \pastebutton{RealClosureXmpPageFull62}{\hidepaste} \tab{5}\spadcommand{f3:Ran:=sqrt(3,5)\free{Ran }\bound{f3 }} \indentrel{3}\begin{verbatim} 5�Ŀ (62) \�3 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty62} \begin{paste}{RealClosureXmpPageEmpty62}{RealClosureXmpPagePatch62} \pastebutton{RealClosureXmpPageEmpty62}{\showpaste} \tab{5}\spadcommand{f3:Ran:=sqrt(3,5)\free{Ran }\bound{f3 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch63} \begin{paste}{RealClosureXmpPageFull63}{RealClosureXmpPageEmpty63} \pastebutton{RealClosureXmpPageFull63}{\hidepaste} \tab{5}\spadcommand{f25:Ran:=sqrt(1/25,5)\free{Ran }\bound{f25 }} \indentrel{3}\begin{verbatim} ��Ŀ � 1 (63) 5��� \�25 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty63} \begin{paste}{RealClosureXmpPageEmpty63}{RealClosureXmpPagePatch63} \pastebutton{RealClosureXmpPageEmpty63}{\showpaste} \tab{5}\spadcommand{f25:Ran:=sqrt(1/25,5)\free{Ran }\bound{f25 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch64} \begin{paste}{RealClosureXmpPageFull64}{RealClosureXmpPageEmpty64} \pastebutton{RealClosureXmpPageFull64}{\hidepaste} \tab{5}\spadcommand{f32:Ran:=sqrt(32/5,5)\free{Ran }\bound{f32 }} \indentrel{3}\begin{verbatim} ��Ŀ �32 (64) 5��� \� 5 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty64} \begin{paste}{RealClosureXmpPageEmpty64}{RealClosureXmpPagePatch64} \pastebutton{RealClosureXmpPageEmpty64}{\showpaste} \tab{5}\spadcommand{f32:Ran:=sqrt(32/5,5)\free{Ran }\bound{f32 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch65} \begin{paste}{RealClosureXmpPageFull65}{RealClosureXmpPageEmpty65} \pastebutton{RealClosureXmpPageFull65}{\hidepaste} \tab{5}\spadcommand{f27:Ran:=sqrt(27/5,5)\free{Ran }\bound{f27 }} \indentrel{3}\begin{verbatim} ��Ŀ �27 (65) 5��� \� 5 Type: RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty65} \begin{paste}{RealClosureXmpPageEmpty65}{RealClosureXmpPagePatch65} \pastebutton{RealClosureXmpPageEmpty65}{\showpaste} \tab{5}\spadcommand{f27:Ran:=sqrt(27/5,5)\free{Ran }\bound{f27 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch66} \begin{paste}{RealClosureXmpPageFull66}{RealClosureXmpPageEmpty66} \pastebutton{RealClosureXmpPageFull66}{\hidepaste} \tab{5}\spadcommand{dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)\free{f32 }\free{f27 }\free{f25 }\free{f3 }\bound{dst5 }} \indentrel{3}\begin{verbatim} ���������������Ŀ � ��Ŀ ��Ŀ ��Ŀ � �27 �32 5�Ŀ2 5�Ŀ � 1 (66) 3�- 5��� + 5��� = (- \�3 + \�3 + 1) 5��� \� \� 5 \� 5 \�25 Type: Equation RealClosure Fraction Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty66} \begin{paste}{RealClosureXmpPageEmpty66}{RealClosureXmpPagePatch66} \pastebutton{RealClosureXmpPageEmpty66}{\showpaste} \tab{5}\spadcommand{dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)\free{f32 }\free{f27 }\free{f25 }\free{f3 }\bound{dst5 }} \end{paste}\end{patch} \begin{patch}{RealClosureXmpPagePatch67} \begin{paste}{RealClosureXmpPageFull67}{RealClosureXmpPageEmpty67} \pastebutton{RealClosureXmpPageFull67}{\hidepaste} \tab{5}\spadcommand{dst5::Boolean\free{dst5 }} \indentrel{3}\begin{verbatim} (67) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RealClosureXmpPageEmpty67} \begin{paste}{RealClosureXmpPageEmpty67}{RealClosureXmpPagePatch67} \pastebutton{RealClosureXmpPageEmpty67}{\showpaste} \tab{5}\spadcommand{dst5::Boolean\free{dst5 }} \end{paste}\end{patch}