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

<b>let </b><font color="Maroon">p1</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>, <font color="Maroon">p2</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>, <font color="Maroon">p3</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>, <font color="Maroon">p4</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">P</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>set </b><font color="Maroon">K</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">E29</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="jgraph_6.html#K5" target="_self">circle</a> 0,0,1 &amp; <font color="Maroon">f</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">E30</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/>;<br/>
<a NAME="E3:95"/><i><font color="Green">E31</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:95"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E10"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:95"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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">p2</font>,<font color="Maroon">p3</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">p3</font>,<font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</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">E36</font></i>: 
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</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">p2</font>,<font color="Maroon">p3</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">p3</font>,<font color="Maroon">p4</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">E37</font></i>: 
( <font color="Maroon">p1</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">p2</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="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<a NAME="E3:95_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">p3</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">p4</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="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">q8</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">E76</font></i>: 
( <font color="Maroon">q8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> &amp; ( ( <font color="Maroon">q8</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">q8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">q8</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">q8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">q8</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">q8</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">q8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">q8</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">q8</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 <b>by </b><i/>;<br/>
<a NAME="E6:95_1"/><i><font color="Green">E77</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</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="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/>
<a NAME="E7:95_1"/><i><font color="Green">E78</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p4</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="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/>
<a NAME="E8:95_1"/><i><font color="Green">E79</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/>;<br/>
<a NAME="E9:95_1"/><i><font color="Green">E92</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">E93</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">p1</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">p1</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">p1</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">p1</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">p1</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#E5:95_1"><i><font color="Green">E76</font></i></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">E96</font></i>: 
<font color="Maroon">p1</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">E98</font></i>: 
( <font color="Maroon">p1</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">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E3:95_1_2_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E5:95_1_2_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E101</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E102</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E7:95_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">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">p1</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">E103</font></i>: 
<font color="Maroon">p1</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">E122</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_1"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E3:95_1_2_1_1_1_1_1"/><i><font color="Green">E124</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E4:95_1_2_1_1_1_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <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">E125</font></i>: 
<font color="Maroon">p1</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">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p2</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">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p2</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">E158</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p2</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">E159</font></i>: 
( <font color="Maroon">p2</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">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E7:95_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/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></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">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p3</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">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p3</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">E164</font></i>: 
( <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p3</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">E165</font></i>: 
( <font color="Maroon">p3</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">p3</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_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/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , </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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p4</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">E169</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p4</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">E171</font></i>: 
( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><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">E172</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E173</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E175</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">E177</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">E179</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , , , <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">E180</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E7:95_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/><a NAME="E11:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , , , <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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <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">E181</font></i>: 
(  not <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p4</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">E182</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">p4</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">p4</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">p4</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">p4</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#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1"><i><font color="Green">E76</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_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">p4</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p4</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">p4</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">p4</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">p4</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="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">p4</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p4</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">p4</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">p4</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">p4</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">p4</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p4</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">p4</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">p4</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">p4</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">E184</font></i>: 
<font color="Maroon">p4</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">E206</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/>;<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">E207</font></i>: 
<font color="Maroon">p4</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">p4</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="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="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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p4</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">p4</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">p4</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">p4</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/>;<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">E208</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E209</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">E210</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">E211</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , , , <a class="ref" href="jgraph_5.html#T54">JGRAPH_5:54</a></i>;<br/><div><i><font color="Green">E212</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  ) or  not <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">E213</font></i>: 
( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p4</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">p4</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">p4</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">p4</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/>;<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">p4</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">p4</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="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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><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/>;<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">p4</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>, </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">E214</font></i>: 
( <font color="Maroon">p4</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">p4</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">E215</font></i>: 
<font color="Maroon">p4</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/>;<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/>;<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="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">E216</font></i>: 
(  not <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">E217</font></i>: 
( <font color="Maroon">p4</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">p4</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">p4</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">p4</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#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1"><i><font color="Green">E76</font></i></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">p4</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  ) )
 </span><b>by </b><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">E218</font></i>: 
<font color="Maroon">p4</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E219</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E220</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E221</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E222</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E223</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_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="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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E224</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T21" target="_self">Th21</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">E225</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E226</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="jgraph_6.html#T22" target="_self">Th22</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">p4</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">p4</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">p4</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>, , </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">p4</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">E227</font></i>: 
<font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><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">E228</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</font></i></a>, , <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E28"><i><font color="Green">Lemma133</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <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">p4</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">E229</font></i>: 
<font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E230</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E231</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E232</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E28"><i><font color="Green">Lemma133</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">E233</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E28"><i><font color="Green">Lemma133</font></i></a>, <a class="txt" href="jgraph_6.html#E29"><i><font color="Green">Lemma134</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">E234</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E235</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E236</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E29"><i><font color="Green">Lemma134</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E237</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">E238</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">E239</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E240</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E298</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E300</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E29"><i><font color="Green">Lemma134</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E301</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E30"><i><font color="Green">Lemma135</font></i></a>, , <a class="ref" href="jgraph_6.html#T23" target="_self">Th23</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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E31"><i><font color="Green">Lemma136</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <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">E302</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E30"><i><font color="Green">Lemma135</font></i></a>, <a class="txt" href="jgraph_6.html#E31"><i><font color="Green">Lemma136</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E303</font></i>: 
<font color="Maroon">p4</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">p4</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">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><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">E304</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E306</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</font></i></a>, , <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_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="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">E307</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="txt" href="jgraph_6.html#E28"><i><font color="Green">Lemma133</font></i></a>, <a class="txt" href="jgraph_6.html#E29"><i><font color="Green">Lemma134</font></i></a>, <a class="txt" href="jgraph_6.html#E30"><i><font color="Green">Lemma135</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E308</font></i>: 
( <font color="Maroon">p4</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">p4</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E309</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E310</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E311</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E313</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_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">E314</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E315</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E316</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E317</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E318</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E125</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E319</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T21" target="_self">Th21</a>, , <a class="ref" href="jgraph_6.html#T22" target="_self">Th22</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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E28"><i><font color="Green">Lemma133</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">E320</font></i>: 
( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p4</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">E321</font></i>: 
<font color="Maroon">p4</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#E11:95_1"><i><font color="Green">E93</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_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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E322</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E323</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E324</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_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_3"/><i><font color="Green">E325</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="jgraph_6.html#T21" target="_self">Th21</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">E326</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E327</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">E328</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , , <a class="ref" href="jgraph_6.html#T22" target="_self">Th22</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E329</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">x</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">x</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">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"/><i><font color="Green">E330</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 <b>by </b><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">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">q</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">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">x</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">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"/><i><font color="Green">E331</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">x</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>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</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">E332</font></i>: 
(  not <font color="Maroon">p3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p3</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">E333</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">p3</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">p3</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">p3</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">p3</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#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1"><i><font color="Green">E76</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_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">p3</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">p3</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">p3</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p3</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">p3</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">p3</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">p3</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="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">p3</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">p3</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">p3</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p3</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">p3</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">p3</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">p3</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">p3</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">p3</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">p3</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p3</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">p3</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">p3</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">p3</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">E334</font></i>: 
<font color="Maroon">p3</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">E335</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/>;<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">E336</font></i>: 
<font color="Maroon">p3</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">p3</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="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="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">p3</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">p3</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p3</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">p3</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">p3</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">p3</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/>;<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">E337</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">p3</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_1_2_1_1"><i><font color="Green">E98</font></i></a>, </i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green">E338</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">E339</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E6:95_1_2_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</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">E340</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E98</font></i></a>, , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><div><i><font color="Green">E341</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  ) or  not <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">E342</font></i>: 
( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">E343</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">E344</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">E345</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">E373</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">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</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#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E35</font></i></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">E374</font></i>: 
<font color="Maroon">p4</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="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">p3</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">p3</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p3</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">p3</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">p3</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">p3</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/>;<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">E375</font></i>: 
( <font color="Maroon">p3</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">p3</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">p3</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>, , , , </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">E376</font></i>: 
<font color="Maroon">p3</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">p4</font>,<font color="Maroon">p3</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>, </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">E377</font></i>: 
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 <b>by </b><i>, , <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">p3</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </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>, </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">E378</font></i>: 
<font color="Maroon">p3</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">p4</font>,<font color="Maroon">p3</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>, </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">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 <b>by </b><i>, , <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>, , </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">E379</font></i>: 
( <font color="Maroon">p3</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">p3</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">p4</font>,<font color="Maroon">p3</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>, </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">E380</font></i>: 
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 <b>by </b><i>, , <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">E381</font></i>: 
<font color="Maroon">p3</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>, </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/>;<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"><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">E382</font></i>: 
(  not <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">E383</font></i>: 
( <font color="Maroon">p4</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">p4</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">p4</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">p4</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#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1"><i><font color="Green">E76</font></i></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">p4</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  ) )
 </span><b>by </b><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">E384</font></i>: 
<font color="Maroon">p4</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E385</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E386</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E387</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E388</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E389</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E390</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, </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">E391</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E392</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="jgraph_6.html#T21" target="_self">Th21</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">p4</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">p4</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">p4</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>, , </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">p4</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">E393</font></i>: 
<font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><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">E394</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E125</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">p4</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">E395</font></i>: 
<font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E396</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E397</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E398</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E399</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="jgraph_6.html#T22" target="_self">Th22</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">E400</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E401</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E402</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="jgraph_6.html#T22" target="_self">Th22</a>, <a class="txt" href="jgraph_6.html#E29"><i><font color="Green">Lemma134</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E403</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">E404</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">E405</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E406</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E407</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E408</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="jgraph_6.html#T22" target="_self">Th22</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E409</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, , </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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <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">E410</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E28"><i><font color="Green">Lemma133</font></i></a>, <a class="txt" href="jgraph_6.html#E31"><i><font color="Green">Lemma136</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="txt" href="jgraph_6.html#E28"><i><font color="Green">Lemma133</font></i></a>, <a class="txt" href="jgraph_6.html#E30"><i><font color="Green">Lemma135</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E411</font></i>: 
<font color="Maroon">p4</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">p4</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">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><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">E412</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E413</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E414</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , , <a class="ref" href="jgraph_6.html#T22" target="_self">Th22</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E415</font></i>: 
( <font color="Maroon">p4</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">p4</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E416</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E417</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E418</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E419</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">E420</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E421</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E422</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E423</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E424</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E425</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="jgraph_6.html#T21" target="_self">Th21</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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <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">E426</font></i>: 
( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p4</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">E427</font></i>: 
<font color="Maroon">p4</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#E11:95_1"><i><font color="Green">E93</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_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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E428</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E429</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E430</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E431</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">E432</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E433</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">E434</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="jgraph_6.html#T21" target="_self">Th21</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E435</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">x</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">x</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">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"/><i><font color="Green">E436</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 <b>by </b><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">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">q</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">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">x</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">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"/><i><font color="Green">E437</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">x</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>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</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">E438</font></i>: 
(  not <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p2</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">E439</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">p2</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">p2</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">p2</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">p2</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#E5:95_1"><i><font color="Green">E76</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_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">p2</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">p2</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">p2</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p2</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">p2</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">p2</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">p2</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="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">p2</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">p2</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">p2</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p2</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">p2</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">p2</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">p2</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">p2</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">p2</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">p2</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p2</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">p2</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">p2</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">p2</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">E440</font></i>: 
<font color="Maroon">p2</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">E441</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/>;<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">E442</font></i>: 
<font color="Maroon">p2</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">p2</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="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="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">p2</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">p2</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p2</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">p2</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">p2</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">p2</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/>;<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">E443</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">p2</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_1_2_1_1"><i><font color="Green">E98</font></i></a>, </i>;<br/><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E444</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></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">p3</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="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">E445</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="txt" href="jgraph_6.html#E2:95_1_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><div><i><font color="Green">E446</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  ) or  not <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">E447</font></i>: 
( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">E448</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">E449</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">E450</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">E451</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">p4</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p4</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#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E4:95"><i><font color="Green">E35</font></i></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">E452</font></i>: 
<font color="Maroon">p4</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="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">p2</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">p2</font> <a href="euclid.html#K22">`2</a>  ) or <font color="Maroon">p2</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">p2</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">p2</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">p2</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/>;<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">E453</font></i>: 
( <font color="Maroon">p2</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">p2</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">p2</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>, , , , </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">E454</font></i>: 
<font color="Maroon">p2</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">p4</font>,<font color="Maroon">p2</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>, </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">E455</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 <b>by </b><i>, , <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">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </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>, </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">E456</font></i>: 
<font color="Maroon">p2</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">p4</font>,<font color="Maroon">p2</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>, </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">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 <b>by </b><i>, , <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>, , </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">E457</font></i>: 
( <font color="Maroon">p2</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">p2</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">p4</font>,<font color="Maroon">p2</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>, </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">E458</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 <b>by </b><i>, , <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">E459</font></i>: 
<font color="Maroon">p2</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>, </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/>;<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="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">E460</font></i>: 
(  not <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or  not <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p4</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">E461</font></i>: 
( <font color="Maroon">p4</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">p4</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">p4</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">p4</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#E9:95_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jgraph_6.html#E5:95_1"><i><font color="Green">E76</font></i></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">p4</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">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  ) )
 </span><b>by </b><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">E462</font></i>: 
<font color="Maroon">p4</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E463</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E464</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E465</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E466</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E467</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E468</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, </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">E469</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E470</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">p4</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">p4</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">p4</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>, , </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">p4</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">E471</font></i>: 
<font color="Maroon">p4</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><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">E472</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <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">p4</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">E473</font></i>: 
<font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E474</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E475</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E476</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E477</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_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="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">E478</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E479</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E480</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E125</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E481</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">E482</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">E483</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E484</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E485</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E486</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jgraph_6.html#E30"><i><font color="Green">Lemma135</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E487</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T21" target="_self">Th21</a>, , <a class="txt" href="jgraph_6.html#E29"><i><font color="Green">Lemma134</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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E28"><i><font color="Green">Lemma133</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E31"><i><font color="Green">Lemma136</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">E488</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jgraph_6.html#T21" target="_self">Th21</a>, , , <a class="ref" href="jgraph_6.html#T22" target="_self">Th22</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E489</font></i>: 
<font color="Maroon">p4</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">p4</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">p4</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><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">E490</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E491</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E492</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , , <a class="txt" href="jgraph_6.html#E1:95_1_2_1_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="jgraph_6.html#T21" target="_self">Th21</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E493</font></i>: 
( <font color="Maroon">p4</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">p4</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E494</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E495</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E496</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E497</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">E498</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E499</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E500</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">E501</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="jgraph_5.html#T32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green">E502</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E503</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, , </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">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <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">E504</font></i>: 
( <font color="Maroon">p4</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">p4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p4</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">E505</font></i>: 
<font color="Maroon">p4</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#E11:95_1"><i><font color="Green">E93</font></i></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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">E506</font></i>: 
<font color="Maroon">f</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">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</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">E507</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">E508</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:95_1"><i><font color="Green">E92</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">E509</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">E510</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E5:95_1_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</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">E511</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E3:95_1_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, , , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</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">E512</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</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="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , , , , , <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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">P</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">E513</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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="ref" href="jgraph_6.html#T6" target="_self">Th6</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">x</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">x</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">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_3_1"/><i><font color="Green">E514</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:95_1_2_1_1_1_1_2_1_1_2_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">q</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">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or (  <a href="real_1.html#K1">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) ) </span>}</span> 
 <b>by </b><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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">x</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">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Olive">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</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">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:95_1_2_1_1_1_1_2_1_1_2_3_2"/><i><font color="Green">E515</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 ) or ( <font color="Maroon">p</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">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">p</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">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</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">x</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>, </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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</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">E516</font></i>: 
<font color="Maroon">p1</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">E517</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">E518</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">E519</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">p1</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">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E6:95_1_2_1_2"/><b>then </b><i><font color="Green">E520</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">p1</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>, , , </i>;<br/><a NAME="E7:95_1_2_1_2"/><b>then </b><i><font color="Green">E521</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E8:95_1_2_1_2"/><i><font color="Green">E522</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">p2</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="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">E523</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></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">p3</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="ref" href="jgraph_6.html#T20" target="_self">Th20</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><b>hence </b><a NAME="E12:95_1_2_1_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T18" target="_self">Th18</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">E524</font></i>: 
<font color="Maroon">p1</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">E525</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">E526</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">E527</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">E528</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">E529</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">E530</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">E531</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="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, </i>;<br/><a NAME="E9:95_1_2_1_3"/>
( <font color="Maroon">p1</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">p1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, </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">p1</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="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E11:95_1_2_1_3"/><b>then </b><i><font color="Green">E532</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">p1</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="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">E533</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E13:95_1_2_1_3"/><i><font color="Green">E534</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">p2</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="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">E535</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></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">p3</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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><b>hence </b><a NAME="E17:95_1_2_1_3"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <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">E536</font></i>: 
( <font color="Maroon">p1</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">p1</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">E537</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">E538</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">E539</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">E540</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">E541</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>, , , </i>;<br/><a NAME="E8:95_1_2_1_4"/><i><font color="Green">E542</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">E543</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">E544</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="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, </i>;<br/><a NAME="E12:95_1_2_1_4"/><b>then </b><i><font color="Green">E545</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="ref" href="jgraph_6.html#T18" target="_self">Th18</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/>;<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">E546</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">p1</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">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, </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">p1</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>, , , , </i>;<br/><a NAME="E18:95_1_2_1_4"/><b>then </b><i><font color="Green">E547</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">p1</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="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">E548</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E20:95_1_2_1_4"/><i><font color="Green">E549</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">p2</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="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">E550</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></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">p3</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="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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="txt" href="jgraph_6.html#E8:95_1"><i><font color="Green">E79</font></i></a></i>;<br/><b>hence </b><a NAME="E24:95_1_2_1_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="jordan17.html#R1">are_in_this_order_on</a> <font color="Maroon">P</font>
 ;<br/>


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


</div>
