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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <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>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</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>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">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>;<br/>





<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1;<br/>
<b>assume </b><a NAME="E1:95"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="jgraph_6.html#K5" target="_self">circle</a> 0,0,1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a>  <a href="jgraph_3.html#K1">Sq_Circ</a>  )
 ;<br/>

<a NAME="E2:95"/><i><font color="Green">E87</font></i>: 
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E3:95"/><i><font color="Green">E88</font></i>: 
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<a NAME="E4:95"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="ref" href="jgraph_6.html#T33" target="_self">Th33</a></i>;<br/>
<b>thus </b><a NAME="E5:95"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_1"/><i><font color="Green">E90</font></i>: 
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 )
 ;<br/>

<a NAME="E2:95_1"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<a NAME="E3:95_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></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="E5:95_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( ( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</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">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>8</sub></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_6.html#E3:95"><i><font color="Green">E88</font></i></a></i>;<br/>
<a NAME="E6:95_1"/><i><font color="Green">E94</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/>
<a NAME="E7:95_1"/><i><font color="Green">E95</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/>
<a NAME="E8:95_1"/><i><font color="Green">E96</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E9:95_1"/><i><font color="Green">E97</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,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/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T36">EUCLID:36</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T62">EUCLID:62</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T62">EUCLID:62</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T60">EUCLID:60</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>

;<br/>
<a NAME="E11:95_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="goboard7.html#T7">GOBOARD7:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self">Th73</a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:95_1_2_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self">Th78</a></i>;<br/><a NAME="E4:95_1_2_1_1"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E5:95_1_2_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E6:95_1_2_1_1"/><i><font color="Green">E103</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:95_1_2_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>1</sub></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="E2:95_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T75" target="_self">Th75</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_1"/><i><font color="Green">E107</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="jordan17.html#D1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> ) or  not <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self">Th65</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> ) or  not <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self">Th65</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> ) or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E116</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E117</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E118</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E120</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self">Th65</a></i>;<br/><a NAME="E12:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E13:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/><b>hence </b><a NAME="E14:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jordan17.html#D1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E114</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><div><i><font color="Green">E115</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self">Th73</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E114</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="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_2"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_3"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E117</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self">Th56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green">E116</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E116</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E117</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/><div><i><font color="Green">E120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E115</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green">E121</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="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green">E121</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green">E123</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"/><i><font color="Green">E121</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self">Th73</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 or ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"><i><font color="Green">E121</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E125</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E126</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E127</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E128</font></i></a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E131</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</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="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</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_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green">E132</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green">E133</font></i></a></i>;<br/><b>hence </b><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E134</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E137</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self">Th78</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E142</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E143</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E145</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E141</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E143</font></i></a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E142</font></i></a></i>;<br/><b>hence </b><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E145</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E140</font></i></a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E141</font></i></a>, <a class="ref" href="jgraph_5.html#T59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>4</sub></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_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E135</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E132</font></i></a>, <a class="ref" href="jgraph_6.html#T80" target="_self">Th80</a></i>;<br/><b>hence </b><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E135</font></i></a>, <a class="ref" href="jgraph_5.html#T59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></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="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E125</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</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_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E129</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E130</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E132</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E12:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E130</font></i></a></i>;<br/><a NAME="E13:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E14:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E129</font></i></a></i>;<br/><b>hence </b><a NAME="E15:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jgraph_6.html#E12:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="jgraph_6.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E126</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E129</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E124</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self">Th78</a></i>;<br/><b>thus </b><a NAME="E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E131</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E121</font></i>: 
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><b>thus </b><a NAME="E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></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="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></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_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E121</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"><i><font color="Green">E122</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></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="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"><i><font color="Green">E122</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>thus </b><a NAME="E12:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jordan17.html#D1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E110</font></i>: 
(  not <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self">Th73</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E110</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="95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_2"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_3"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E113</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self">Th56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"><i><font color="Green">E113</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"><i><font color="Green">E112</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:95_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E113</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E115</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><div><i><font color="Green">E116</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_6.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E111</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self">Th65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><i><font color="Green">E124</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green">E125</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"/><i><font color="Green">E117</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self">Th73</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 or ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"><i><font color="Green">E117</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E121</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E122</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E123</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E124</font></i></a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E127</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</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="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</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_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green">E128</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green">E129</font></i></a></i>;<br/><b>hence </b><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E130</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E133</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E135</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E11:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self">Th78</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E138</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E139</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E140</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E139</font></i></a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E138</font></i></a></i>;<br/><b>hence </b><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E141</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E142</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E136</font></i></a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E137</font></i></a>, <a class="ref" href="jgraph_5.html#T59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>4</sub></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_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E131</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E128</font></i></a>, <a class="ref" href="jgraph_6.html#T80" target="_self">Th80</a></i>;<br/><b>hence </b><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E131</font></i></a>, <a class="ref" href="jgraph_5.html#T59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></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="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E121</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</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_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E125</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E126</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E128</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E12:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/><a NAME="E13:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E14:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E125</font></i></a></i>;<br/><b>hence </b><a NAME="E15:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jgraph_6.html#E12:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="jgraph_6.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E122</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E122</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E125</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E119</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self">Th78</a></i>;<br/><b>thus </b><a NAME="E11:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green">E127</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E117</font></i>: 
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><b>thus </b><a NAME="E10:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></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="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></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_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E117</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"><i><font color="Green">E118</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></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="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"><i><font color="Green">E118</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>thus </b><a NAME="E11:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="jordan17.html#D1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:95_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2"/><i><font color="Green">E106</font></i>: 
(  not <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><div><i><font color="Green">E107</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self">Th73</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_1_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E106</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="95_1_2_1_1_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_1_1_2"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_1_1_3"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_1_1_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_1_1_4"/><i><font color="Green">E109</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self">Th56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_1_1_4_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2_1_1_4"><i><font color="Green">E109</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2_1_1_4"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_1_1_4_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_1_1_4"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_1_1_4"><i><font color="Green">E108</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:95_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2"/>
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E115</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E116</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_6.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E107</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self">Th65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"><i><font color="Green">E119</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><i><font color="Green">E118</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E7:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><i><font color="Green">E120</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:95_1_2_1_1_1_1_2_1_1_2_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E112</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or  not <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self">Th73</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1 or ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2"><i><font color="Green">E112</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E116</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E117</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E118</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E119</font></i></a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E122</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</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="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</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_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green">E124</font></i></a></i>;<br/><b>hence </b><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E125</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E128</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E130</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E11:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self">Th78</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E133</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E134</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E136</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E134</font></i></a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E133</font></i></a></i>;<br/><b>hence </b><a NAME="E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green">E137</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E131</font></i></a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="jgraph_5.html#T59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>4</sub></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_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green">E126</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jgraph_6.html#T80" target="_self">Th80</a></i>;<br/><b>hence </b><a NAME="E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green">E126</font></i></a>, <a class="ref" href="jgraph_5.html#T59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></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="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E116</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</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_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E120</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E121</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T37">JGRAPH_5:37</a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>6</sub></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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E123</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E12:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E121</font></i></a></i>;<br/><a NAME="E13:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_5.html#T38">JGRAPH_5:38</a></i>;<br/><a NAME="E14:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E120</font></i></a></i>;<br/><b>hence </b><a NAME="E15:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jgraph_6.html#E12:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="jgraph_6.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self">Th45</a>, <a class="ref" href="jgraph_3.html#T33">JGRAPH_3:33</a></i>;<br/><a NAME="E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E117</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E120</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self">Th81</a></i>;<br/><a NAME="E10:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self">Th78</a></i>;<br/><b>thus </b><a NAME="E11:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E8:95_1_2_1_1_1_1_2_1_1_2"/><i><font color="Green">E112</font></i>: 
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E37"><i><font color="Green">Lemma34</font></i></a></i>;<br/><b>thus </b><a NAME="E9:95_1_2_1_1_1_1_2_1_1_2"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_1_2_1_1_1_1_2_1_1_2_3"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:95_1_2_1_1_1_1_2_1_1_2_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></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="E3:95_1_2_1_1_1_1_2_1_1_2_3_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></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_6.html#E8:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E112</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_3_1"><i><font color="Green">E113</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:95_1_2_1_1_1_1_2_1_1_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1_1_1_2_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:95_1_2_1_1_1_1_2_1_1_2_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></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="E3:95_1_2_1_1_1_1_2_1_1_2_3_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( ( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></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">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_3_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1_1_1_2_1_1_2_3_2"><i><font color="Green">E113</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>thus </b><a NAME="E10:95_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_1_1_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="jordan17.html#D1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:95_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:95_1_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_2"><b>case </b></a><a NAME="E1:95_1_2_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_2"/><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E3:95_1_2_1_2"/><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E4:95_1_2_1_2"/><i><font color="Green">E102</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E5:95_1_2_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</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">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>1</sub></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_6.html#E1:95_1_2_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E6:95_1_2_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>1</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E2:95_1_2_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E7:95_1_2_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><a NAME="E8:95_1_2_1_2"/><i><font color="Green">E105</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E9:95_1_2_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><a NAME="E10:95_1_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E11:95_1_2_1_2"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><b>hence </b><a NAME="E12:95_1_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:95_1_2_1_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="jordan17.html#D1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_3"><b>case </b></a><a NAME="E1:95_1_2_1_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_3"/><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E3:95_1_2_1_3"/><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E4:95_1_2_1_3"/><i><font color="Green">E102</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E5:95_1_2_1_3"/><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E6:95_1_2_1_3"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E7:95_1_2_1_3"/><i><font color="Green">E105</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</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">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E8:95_1_2_1_3"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_3"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_3"><i><font color="Green">E104</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E9:95_1_2_1_3"/>
( <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_3"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E10:95_1_2_1_3"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>1</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T71" target="_self">Th71</a></i>;<br/><a NAME="E11:95_1_2_1_3"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>1</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_3"><i><font color="Green">E106</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E12:95_1_2_1_3"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_3"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_3"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><a NAME="E13:95_1_2_1_3"/><i><font color="Green">E109</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E11:95_1_2_1_3"><i><font color="Green">E107</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E14:95_1_2_1_3"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_3"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_3"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><a NAME="E15:95_1_2_1_3"/>
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E13:95_1_2_1_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E16:95_1_2_1_3"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_3"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_3"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><b>hence </b><a NAME="E17:95_1_2_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E12:95_1_2_1_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jgraph_6.html#E14:95_1_2_1_3"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan17.html#D1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_4"><b>case </b></a><a NAME="E1:95_1_2_1_4"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_4"/><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E3:95_1_2_1_4"/><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E4:95_1_2_1_4"/><i><font color="Green">E102</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E5:95_1_2_1_4"/><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E6:95_1_2_1_4"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</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">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E7:95_1_2_1_4"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95_1_2_1_4"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_4"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_4"><i><font color="Green">E103</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E8:95_1_2_1_4"/><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E9:95_1_2_1_4"/><i><font color="Green">E106</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E10:95_1_2_1_4"/><i><font color="Green">E107</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E11:95_1_2_1_4"/>
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:95_1_2_1_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1_2_1_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E12:95_1_2_1_4"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1_2_1_4"><i><font color="Green">E104</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E13:95_1_2_1_4"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E14:95_1_2_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E15:95_1_2_1_4"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E16:95_1_2_1_4"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>1</sub></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_6.html#E1:95_1_2_1_4"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E17:95_1_2_1_4"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>1</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_4"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E9:95_1_2_1_4"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jgraph_6.html#E10:95_1_2_1_4"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jgraph_6.html#E15:95_1_2_1_4"><i><font color="Green">E109</font></i></a>, <a class="ref" href="jgraph_6.html#T72" target="_self">Th72</a></i>;<br/><a NAME="E18:95_1_2_1_4"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>1</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E12:95_1_2_1_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E19:95_1_2_1_4"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_4"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_4"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><a NAME="E20:95_1_2_1_4"/><i><font color="Green">E112</font></i>: 
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>2</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E18:95_1_2_1_4"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E21:95_1_2_1_4"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_4"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_4"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><a NAME="E22:95_1_2_1_4"/>
 <a href="jordan6.html#R1">LE</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:95"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E20:95_1_2_1_4"><i><font color="Green">E112</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><a NAME="E23:95_1_2_1_4"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_4"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95_1_2_1_4"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self">Th76</a></i>;<br/><b>hence </b><a NAME="E24:95_1_2_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E19:95_1_2_1_4"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jgraph_6.html#E21:95_1_2_1_4"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan17.html#D1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:95_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>


</div>
