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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><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>;<br/>



<b>set </b><font color="Maroon">K</font> =  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<b>assume </b><a NAME="E1:81"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> &amp; <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"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &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"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/>

<a NAME="E2:81"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:81"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <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> <font color="Maroon">b</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:81"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:81"/><i><font color="Green">E36</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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:81"/><i><font color="Green">E37</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:81"/><i><font color="Green">E38</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:81"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:81"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:81"/><i><font color="Green">E78</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T76">JORDAN6:76</a></i>;<br/>
<a NAME="E11:81"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<div><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 ;<br/><a NAME="E2:81_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jgraph_6.html#E11:81"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:81_1"/><b>then </b>
( <font color="Maroon">p1</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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:81_1"/>
<font color="Maroon">p1</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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jgraph_6.html#E9:81"><i><font color="Green">E77</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E13:81"/>
(  <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> &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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> implies ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_2"/><i><font color="Green">E93</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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> &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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 ;<br/>

<a NAME="E2:81_2"/><b>then </b><i><font color="Green">E96</font></i>: 
( ( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; not <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> ) or ( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>, <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>, <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>, <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> ) or ( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; not <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>, <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>, <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>, <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:81_2"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></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">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></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"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; ( for <font color="Olive">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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> ) ) &amp; ( for <font color="Olive">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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> ) ) )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">s1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>, <font color="Maroon">s2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:81_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, <a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E7:81_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E8:81_2"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E9:81_2"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E10:81_2"/><i><font color="Green">E102</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E11:81_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E12:81_2"/><b>then </b>
<font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, , <a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , , <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E13:81_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:81_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E15:81_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2
 
;<br/>
<a NAME="E16:81_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span>
 
;<br/>
<a NAME="E17:81_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/>
<a NAME="E18:81_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E19:81_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E20:81_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E21:81_2"/>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_2_1"/><i><font color="Green">E103</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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 ;<br/><a NAME="E2:81_2_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>, <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan7.html#T3">JORDAN7:3</a></i>;<br/><b>hence </b><a NAME="E3:81_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E23:81_2"/>
<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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 ;<br/>


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

<b>thus </b><a NAME="E14:81"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> implies (  <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> &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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_3"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 ;<br/>

<a NAME="E2:81_3"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span><br/> for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 1 <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> &amp; <font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/><font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:81_3_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:81_3_1"/><i><font color="Green">E158</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:81_3_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E4:81_3_1"/><i><font color="Green">E162</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<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#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E6:81_3_1"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:81_3_1"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> =  <a href="pscomp_1.html#K13">proj1</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h2</font> =  <a href="pscomp_1.html#K14">proj2</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E10:81_3_1"/><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>;<br/>
<a NAME="E11:81_3_1"/><b>then </b><i><font color="Green">E165</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span></span>)</span> holds  <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> ) implies <font color="Maroon">h1</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T39">JGRAPH_2:39</a></i>;<br/>
<a NAME="E12:81_3_1"/>
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span></span>)</span> holds  <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> ) implies <font color="Maroon">h2</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i>, <a class="ref" href="jgraph_2.html#T40">JGRAPH_2:40</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <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>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E14:81_3_1"/><i><font color="Green">E166</font></i>: 
( ( for <font color="Olive">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><br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> &amp; <font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">r2</font> ) &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E12:81"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jgraph_2.html#T29">JGRAPH_2:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g1</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E16:81_3_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">k</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E17:81_3_1"/><i><font color="Green">E169</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">s1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s2</font>
 ;<br/><a NAME="E2:81_3_1_1"/><i><font color="Green">E172</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:81_3_1_1"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>then </b><i><font color="Green">E173</font></i>: <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> 0 = 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><a NAME="E5:81_3_1_1"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>then </b><i><font color="Green">E175</font></i>: <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> = 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E7:81_3_1_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>then </b><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> = 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E9:81_3_1_1"/><b>then </b><i><font color="Green">E179</font></i>: 
( <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> )
 <b>by </b><i>, , <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, , , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E10:81_3_1_1"/><i><font color="Green">E180</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E11:81_3_1_1"/><b>then </b><i><font color="Green">E181</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">s2</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">s2</font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E13:81_3_1_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">B</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/><a NAME="E14:81_3_1_1"/><i><font color="Green">E184</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E15:81_3_1_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E16:81_3_1_1"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> 0 is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E17:81_3_1_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E18:81_3_1_1"/>
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">xc</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E20:81_3_1_1"/><i><font color="Green">E209</font></i>: 
( <font color="Maroon">xc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> )
 <b>by </b><i>, , <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, , , , <a class="ref" href="topreal5.html#T11">TOPREAL5:11</a></i>;<br/><a NAME="E21:81_3_1_1"/>
<font color="Maroon">xc</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><b>then reconsider </b><font color="Maroon">rxc</font> = <font color="Maroon">xc</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E23:81_3_1_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">k</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:81_3_1_1_4"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:81_3_1_1_4_1"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">x1</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i/>;<br/>
<i><font color="Green">E212</font></i>: <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/>
<i><font color="Green">E213</font></i>: <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/>
<a NAME="E6:81_3_1_1_4_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:81_3_1_1_4_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">gr1</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">gr2</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_4_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_4_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_3_1_1_4_1_3_1"/>
( ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> ) or ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> ) or ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> ) or ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E10:81"><i><font color="Green">E78</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_4_1_3_1_1"><b>case </b></a><a NAME="E1:81_3_1_1_4_1_3_1_1"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:81_3_1_1_4_1_3_1_1"/><b>then </b><i><font color="Green">E217</font></i>: 
( <font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:81_3_1_1_4_1_3_1_1"/><i><font color="Green">E218</font></i>: 
( <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:81_3_1_1_4_1_3_1_1"/><b>then </b>
<font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , </i>;<br/><a NAME="E5:81_3_1_1_4_1_3_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font>
 <b>by </b><i>, , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E6:81_3_1_1_4_1_3_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E7:81_3_1_1_4_1_3_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_4_1_3_1_2"><b>case </b></a><a NAME="E1:81_3_1_1_4_1_3_1_2"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:81_3_1_1_4_1_3_1_2"/><b>then </b><i><font color="Green">E220</font></i>: 
( <font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:81_3_1_1_4_1_3_1_2"/><i><font color="Green">E221</font></i>: 
( <font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:81_3_1_1_4_1_3_1_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , , , , </i>;<br/><div><i><font color="Green">E223</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_4_1_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1_1_4_1_3_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E2:81_3_1_1_4_1_3_1_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:81_3_1_1_4_1_3_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_4_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1_1_4_1_3_1_2_2"/>
<font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:81_3_1_1_4_1_3_1_2_2"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:81_3_1_1_4_1_3_1_2_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><a NAME="E7:81_3_1_1_4_1_3_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font>
 <b>by </b><i>, , , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E8:81_3_1_1_4_1_3_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E9:81_3_1_1_4_1_3_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_4_1_3_1_3"><b>case </b></a><a NAME="E1:81_3_1_1_4_1_3_1_3"/><i><font color="Green">E224</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:81_3_1_1_4_1_3_1_3"/><b>then </b><i><font color="Green">E225</font></i>: 
( <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:81_3_1_1_4_1_3_1_3"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:81_3_1_1_4_1_3_1_3"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , , , , </i>;<br/><div><i><font color="Green">E228</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_4_1_3_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1_1_4_1_3_1_3_1"/>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E2:81_3_1_1_4_1_3_1_3_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:81_3_1_1_4_1_3_1_3_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1_1_4_1_3_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1_1_4_1_3_1_3_2"/>
<font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:81_3_1_1_4_1_3_1_3_2"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:81_3_1_1_4_1_3_1_3_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><a NAME="E7:81_3_1_1_4_1_3_1_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font>
 <b>by </b><i>, , , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E8:81_3_1_1_4_1_3_1_3"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E9:81_3_1_1_4_1_3_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_3_1_1_4_1_3_1_4"><b>case </b></a><a NAME="E1:81_3_1_1_4_1_3_1_4"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:81_3_1_1_4_1_3_1_4"/><b>then </b><i><font color="Green">E230</font></i>: 
( <font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:81_3_1_1_4_1_3_1_4"/><i><font color="Green">E231</font></i>: 
( <font color="Maroon">gr2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:81_3_1_1_4_1_3_1_4"/><b>then </b>
<font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">gr2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , </i>;<br/><a NAME="E5:81_3_1_1_4_1_3_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gr1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gr1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font>
 <b>by </b><i>, , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><a NAME="E6:81_3_1_1_4_1_3_1_4"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E7:81_3_1_1_4_1_3_1_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:81_3_1_1_4_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:81_3_1_1_4"/>
<font color="Maroon">k</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:81_3_1_1"/><i><font color="Green">E232</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E25:81_3_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E26:81_3_1_1"/><b>then </b>
<font color="Maroon">rxc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E27:81_3_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:81_3_1"/>
<font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E233</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_2"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">p1</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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 ;<br/><a NAME="E2:81_3_2"/><b>then </b>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:81"><i><font color="Green">E76</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:81_3_2"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:81_3_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E5:81_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:81_3"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; not <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>, <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>, <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>, <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="jgraph_6.html#E11:81"><i><font color="Green">E79</font></i></a>, <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:81_3"/>
 <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>

<b>thus </b><a NAME="E6:81_3"/>
<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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a></i>;<br/>


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


</div>
