<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">Kb</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Kb</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">Kb</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span>  &amp; <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 ;<br/>

<b>set </b><font color="Maroon">v</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E2:34"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E3:34"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 
;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Kb</font></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T12" target="_self">Th12</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">Kb</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E5:34"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T12" target="_self">Th12</a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">PP</font> = <font color="Maroon">P</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Kbb</font> = <font color="Maroon">Kb</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jgraph_3.html#T12" target="_self">Th12</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Kbd</font> = <font color="Maroon">Kbb</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Kbb</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">PP</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">b</font> = 1;<br/>
<b>set </b><font color="Maroon">a</font> = 0;<br/>
<b>set </b><font color="Maroon">d</font> = 1;<br/>
<b>set </b><font color="Maroon">c</font> = 0;<br/>
<b>set </b><font color="Maroon">A</font> = 2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span>;<br/>
<b>set </b><font color="Maroon">B</font> = 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">C</font> = 2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span>;<br/>
<b>set </b><font color="Maroon">D</font> = 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E10:34"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:34_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:34_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:34"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">t2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E4:34_3"/>
for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
;<br/>
<b>hence </b><a NAME="E5:34_3"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:34"/>
 ex <font color="Olive">ff</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 
<b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">ff</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:34"/><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/>for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> holds <br/><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> ) )
 ;<br/>
<a NAME="E15:34"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds  <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E16:34"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:34_4"/>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:34_4"/>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jgraph_3.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E18:34"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">ff</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="jgraph_2.html#T53">JGRAPH_2:53</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f11</font> = <font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <a href="topreal1.html#K2">R^2-unit_square</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E60</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f11</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">ff</font></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="topreal1.html#K2">R^2-unit_square</a> 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="topreal1.html#K2">R^2-unit_square</a> 
<b>by </b><i><a class="ref" href="jgraph_3.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E21:34"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">f11</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E22:34"/>
<font color="Maroon">ff</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:34_6"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ff</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ff</font> &amp; <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">x1</font>, <font color="Maroon">p2</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:34_6"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:34_6"/>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:34_6"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/>
<a NAME="E6:34_6"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E7:34_6"/><b>then </b>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> )
 
;<br/>
<a NAME="E8:34_6"/><b>then </b>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> )
 
;<br/>
<a NAME="E9:34_6"/><b>then </b>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  )
 
;<br/>
<b>hence </b><a NAME="E10:34_6"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:34"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">f11</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> ;<br/>
<a NAME="E24:34"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">Kbd</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f11</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_7"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Kbd</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">py</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon">t</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E3:34_7"/><i><font color="Green">E72</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:34_7"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:34_7"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:34_7"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">py</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; ( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 <b>by </b><i><a class="ref" href="jgraph_3.html#T12" target="_self">Th12</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_7_1_1"/>
( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_1_1"><b>case </b></a><a NAME="E1:34_7_1_1_1"/><i><font color="Green">E76</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:34_7_1_1_1"/><b>then </b>
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E3:34_7_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:34_7_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E5:34_7_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E6:34_7_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E7:34_7_1_1_1"/>
2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/><a NAME="E8:34_7_1_1_1"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E9:34_7_1_1_1"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E10:34_7_1_1_1"/><b>then </b>
2 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E11:34_7_1_1_1"/>
( ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i>, <a class="txt" href="jgraph_3.html#E20:34"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_1_2"><b>case </b></a><a NAME="E1:34_7_1_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:34_7_1_1_2"/><b>then </b>
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E3:34_7_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:34_7_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E5:34_7_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E6:34_7_1_1_2"/>
2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/><a NAME="E7:34_7_1_1_2"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E8:34_7_1_1_2"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E9:34_7_1_1_2"/><b>then </b>
2 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E10:34_7_1_1_2"/>
( ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i>, <a class="txt" href="jgraph_3.html#E20:34"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_1_3"><b>case </b></a><a NAME="E1:34_7_1_1_3"/><i><font color="Green">E82</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:34_7_1_1_3"/><b>then </b>
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E3:34_7_1_1_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:34_7_1_1_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E5:34_7_1_1_3"/><b>then </b><i><font color="Green">E83</font></i>: 
0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E6:34_7_1_1_3"/>
2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i/>;<br/><a NAME="E7:34_7_1_1_3"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E8:34_7_1_1_3"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E9:34_7_1_1_3"/><b>then </b>
2 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E10:34_7_1_1_3"/>
( ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i>, <a class="txt" href="jgraph_3.html#E20:34"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_1_4"><b>case </b></a><a NAME="E1:34_7_1_1_4"/><i><font color="Green">E84</font></i>: 
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:34_7_1_1_4"/><b>then </b>
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E3:34_7_1_1_4"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:34_7_1_1_4"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E5:34_7_1_1_4"/><b>then </b><i><font color="Green">E85</font></i>: 
0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E6:34_7_1_1_4"/>
2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i/>;<br/><a NAME="E7:34_7_1_1_4"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E8:34_7_1_1_4"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E9:34_7_1_1_4"/><b>then </b>
2 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E10:34_7_1_1_4"/>
( ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) or ( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i>, <a class="txt" href="jgraph_3.html#E20:34"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E9:34_7"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K2">R^2-unit_square</a> 
 
<b>by </b><i><a class="ref" href="topreal1.html#T20">TOPREAL1:20</a></i>;<br/>
<a NAME="E10:34_7"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E11:34_7"/><b>then </b>
<font color="Maroon">py</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<b>then </b><font color="Maroon">py</font> = 
<font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">f11</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">py</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E13:34_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f11</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:34"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f11</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Kbd</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f11</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_8"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f11</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:34_8"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( ( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) ) )
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T20">TOPREAL1:20</a></i>;<br/>
<i><font color="Green">E109</font></i>: <font color="Maroon">y</font> = 
<font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T13" target="_self">Th13</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">qy</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_8_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_8_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_8_2_1"/>
( ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_8_2_1_1"><b>suppose </b></a><a NAME="E1:34_8_2_1_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:34_8_2_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:34_8_2_1_1"/>
2 <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:34_8_2_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:34_8_2_1_1"/>
2 <a href="real_1.html#K4">*</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:34_8_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E7:34_8_2_1_1"/>
( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jgraph_3.html#E20:34"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_8_2_1_2"><b>suppose </b></a><a NAME="E1:34_8_2_1_2"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:34_8_2_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:34_8_2_1_2"/>
2 <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:34_8_2_1_2"/><b>then </b><i><font color="Green">E118</font></i>: 
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:34_8_2_1_2"/>
2 <a href="real_1.html#K4">*</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:34_8_2_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E7:34_8_2_1_2"/>
( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jgraph_3.html#E20:34"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_8_2_1_3"><b>suppose </b></a><a NAME="E1:34_8_2_1_3"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:34_8_2_1_3"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:34_8_2_1_3"/>
2 <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:34_8_2_1_3"/><b>then </b><i><font color="Green">E121</font></i>: 
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:34_8_2_1_3"/>
2 <a href="real_1.html#K4">*</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:34_8_2_1_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E7:34_8_2_1_3"/>
( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jgraph_3.html#E20:34"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_8_2_1_4"><b>suppose </b></a><a NAME="E1:34_8_2_1_4"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">t</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:34_8_2_1_4"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:34_8_2_1_4"/>
2 <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:34_8_2_1_4"/><b>then </b><i><font color="Green">E124</font></i>: 
0 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:34_8_2_1_4"/>
2 <a href="real_1.html#K4">*</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:34_8_2_1_4"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E7:34_8_2_1_4"/>
( (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">qy</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">qy</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E15:34"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jgraph_3.html#E18:34"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jgraph_3.html#E20:34"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:34_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Kbd</font>
 <b>by </b><i><a class="ref" href="jgraph_3.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:34"/><b>then </b>
<font color="Maroon">Kbd</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f11</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Kbd</font></span>)</span><b> such that </b><br/><a NAME="E28:34"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">f11</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T64">JGRAPH_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f22</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Kbb</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Kbb</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">PP</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f22</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">PP</font></span>)</span> ;<br/>
<a NAME="E32:34"/>
<font color="Maroon">h</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T12" target="_self">Th12</a>, , <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<b>hence </b><a NAME="E33:34"/>
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 <b>by </b><i><a class="ref" href="topreal2.html#D1">TOPREAL2:def 1</a></i>;<br/>


</div>
