\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: Type
\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}
           +---+    +--+    +---+    +--+
   (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)
                +---+          +--+  +--+
         (54602\|548  + 149602\|73 )\|60
       + 
               +--+ +---+
         49502\|73 \|548  + 9900895
    *
        +---+
       \|586
   + 
             +--+ +---+             +--+            +---+
     (154702\|73 \|548  + 30941947)\|60  + 10238421\|548
   + 
              +--+
     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}
           +---+    +--+    +---+    +---+
   (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)
                 +---+           +---+  +--+
         (556778\|778  + 1209010\|165 )\|86
       + 
                +---+ +---+
         401966\|165 \|778  + 144019431
    *
        +---+
       \|990
   + 
              +---+ +---+              +--+
     (1363822\|165 \|778  + 488640503)\|86
   + 
               +---+             +---+
     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}
           +---+    +---+    +---+    +---+
   (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)
                  +---+         +---+  +---+
         (- 34102\|708  - 61598\|217 )\|226
       + 
                 +---+ +---+
         - 34802\|217 \|708  - 13641141
    *
        +---+
       \|692
   + 
              +---+ +---+             +---+
     (- 60898\|217 \|708  - 23869841)\|226
   + 
                +---+            +---+
     - 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}
            +---+    +---+    +---+    +---+
   (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)
                  +---+         +---+  +---+
         (- 37078\|836  - 86110\|155 )\|162
       + 
                 +---+ +---+
         - 37906\|155 \|836  - 13645107
    *
        +---+
       \|820
   + 
              +---+ +---+             +---+
     (- 85282\|155 \|836  - 30699151)\|162
   + 
                +---+            +---+
     - 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}
            +---+    +---+    +---+    +---+
   (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)
                +---+         +---+  +---+
         (70922\|772  + 81058\|591 )\|552
       + 
               +---+ +---+
         68542\|591 \|772  + 46297673
    *
        +---+
       \|818
   + 
            +---+ +---+             +---+            +---+
     (83438\|591 \|772  + 56359389)\|552  + 47657051\|772
   + 
              +---+
     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}
            +----+    +---+    +----+    +---+
   (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)
                 +----+          +---+  +---+
         (115442\|1053  + 179818\|434 )\|412
       + 
                +---+ +----+
         112478\|434 \|1053  + 76037291
    *
        +----+
       \|1088
   + 
             +---+ +----+              +---+
     (182782\|434 \|1053  + 123564147)\|412
   + 
              +----+             +---+
     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}
            +----+    +---+    +----+    +---+
   (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)
                 +----+          +---+  +---+
         (349522\|1049  + 499322\|514 )\|446
       + 
                +---+ +----+
         325582\|514 \|1049  + 239072537
    *
        +----+
       \|1152
   + 
             +---+ +----+              +---+
     (523262\|514 \|1049  + 384227549)\|446
   + 
               +----+             +---+
     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}
            +----+    +---+    +----+    +---+
   (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)
                   +----+          +---+  +---+
         (- 214702\|1751  - 651782\|190 )\|208
       + 
                  +---+ +----+
         - 224642\|190 \|1751  - 129571901
    *
        +----+
       \|1698
   + 
               +---+ +----+              +---+
     (- 641842\|190 \|1751  - 370209881)\|208
   + 
                 +----+             +---+
     - 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)  [%A33,%A34,%A35,%A36]
             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}
          +--+
   (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)
      +---------+ +--------+    +---------+ +--------+
      | +--+      | +-+         | +--+      | +-+
   - \|\|10  - 3 \|\|5  - 2  + \|\|10  + 3 \|\|5  + 2  =
      +-----------+
      |   +-+
     \|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)
      +--------+ +--------+    +--------+ +--------+
      | +-+      | +-+         | +-+      | +-+
   - \|\|5  - 2 \|\|2  - 1  + \|\|5  + 2 \|\|2  + 1  =
      +----------+
      |  +--+
     \|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}
          +-+
   (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}
          +-+
   (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}
          +-------------+
         3|  +-+     +-+
   (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}
          +-------------+
         3|  +-+     +-+
   (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}
          +----+
   (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}
            +--------------+
         1 3|  +----+
   (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}
              +--------------+
           1 3|  +----+
   (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}
          +---+
   (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}
          +-----------+
         5| +---+
   (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}
            +-----------+
           5| +---+
   (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}
          +---------+
          |  +-+         +-+
   (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}
          +-+
   (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}
          +-----------+    +---------+
          |    +-+         |  +-+         +-+
   (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}
          +--+
   (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)
    +--------------------------------+
    |  +--------------+
    |  |     +--+           +--+
   \|2\|- 10\|29  + 55  - 2\|29  + 16  =
        +-----------+    +----------+
        |  +--+          |  +-+          +-+
     - \|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)
    +--------------------------------+
    |    +-+       +-+      +-+
   \|(34\|2  + 46)\|5  + 70\|2  + 112  =
       +-+      +-+     +-+
     (\|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+-+
   (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}
          +--+
          | 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}
          +--+
          |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}
          +--+
          |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}
          +---------------+
          |   +--+    +--+                         +--+
          |   |27     |32       5+-+2   5+-+       | 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}

