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

<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E3:36"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8">JORDAN6:def 8</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green">E33</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">P4</font> =  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>;<br/>
<a NAME="E5:36"/><i><font color="Green">E34</font></i>: 
(  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></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> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green">E45</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green">E46</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P1</font> =  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">P2</font> =  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">Q</font> =  <a href="jordan6.html#K6">Vertical_Line</a> 0 as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon">p11</font> =  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>;<br/>
<b>set </b><font color="Maroon">p22</font> =  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>;<br/>
<b>set </b><font color="Maroon">pj</font> =  <a href="jordan5c.html#K1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span>;<br/>
<b>set </b><font color="Maroon">p8</font> =  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span>;<br/>
<a NAME="E11:36"/><i><font color="Green">E47</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E12:36"/><i><font color="Green">E48</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E13:36"/><i><font color="Green">E49</font></i>: 
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E14:36"/><i><font color="Green">E73</font></i>: 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_5.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E15:36"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , , , <a class="ref" href="jordan6.html#T85">JORDAN6:85</a></i>;<br/>
<a NAME="E16:36"/><i><font color="Green">E77</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/>

<a NAME="E2:36_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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">1 <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">l</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:36_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</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> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#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">l</font> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
0

;<br/>
<b>hence </b><a NAME="E6:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E12:36"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:36"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:36"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a></i>;<br/>
<a NAME="E18:36"/><i><font color="Green">E80</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T12">JORDAN6:12</a></i>;<br/>
<a NAME="E19:36"/>
 <a href="jordan6.html#K6">Vertical_Line</a> 0 is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T33">JORDAN6:33</a></i>;<br/>
<a NAME="E20:36"/><b>then </b>
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:36"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E21:36"/><b>then </b><i><font color="Green">E82</font></i>: 
(  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> &amp; ( 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> <font color="Maroon">P1</font></span>)</span><br/> for <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> <font color="Maroon">P</font> &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> <font color="Maroon">P</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> &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/>for <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a>  st 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">s2</font> holds <br/> not <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> ) )
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E12:36"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan5c.html#D2">JORDAN5C:def 2</a></i>;<br/>
<a NAME="E22:36"/>
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:36_2"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:36_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 )
 ;<br/>
<a NAME="E5:36_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:36_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E8:36_2"/>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a>, , <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<a NAME="E9:36_2"/><b>then </b>
( <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i><a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/>
<a NAME="E10:36_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a>, , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<b>hence </b><a NAME="E11:36_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:36"/><b>then </b>
(  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E24:36"/><b>then </b><i><font color="Green">E86</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E25:36"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , , , <a class="txt" href="jgraph_5.html#E6:36"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan6.html#T84">JORDAN6:84</a></i>;<br/>
<a NAME="E26:36"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">P2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E11:36"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a></i>;<br/>
<a NAME="E27:36"/><i><font color="Green">E89</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T12">JORDAN6:12</a></i>;<br/>
<a NAME="E28:36"/>
 <a href="jordan6.html#K6">Vertical_Line</a> 0 is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T33">JORDAN6:33</a></i>;<br/>
<a NAME="E29:36"/><b>then </b>
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E30:36"/><b>then </b><i><font color="Green">E90</font></i>: 
(  <a href="jordan5c.html#K1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> &amp; ( 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> <font color="Maroon">P2</font></span>)</span><br/> for <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#K27">W-min</a> <font color="Maroon">P</font> &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#K31">E-max</a> <font color="Maroon">P</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> &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/>for <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s2</font> holds <br/> not <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> ) )
 
<b>by </b><i>, <a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan5c.html#D1">JORDAN5C:def 1</a></i>;<br/>
<a NAME="E31:36"/>
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:36_3"/><b>then </b><i><font color="Green">E94</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:36_3"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 )
 ;<br/>
<a NAME="E5:36_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:36_3"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E8:36_3"/>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<a NAME="E9:36_3"/><b>then </b>
( <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i><a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/>
<a NAME="E10:36_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<b>hence </b><a NAME="E11:36_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:36"/><b>then </b>
(  <a href="jordan5c.html#K1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan5c.html#K1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E33:36"/><b>then </b><i><font color="Green">E98</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 or <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E34:36"/><i><font color="Green">E99</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E35:36"/><i><font color="Green">E100</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E36:36"/><i><font color="Green">E101</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E37:36"/>
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</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> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E38:36"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E39:36"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</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> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E40:36"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R</font> =  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<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> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E43:36"/><i><font color="Green">E105</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> <font color="Maroon">P</font> &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> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>then consider </b><font color="Maroon">x8</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E46:36"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x8</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x8</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E47:36"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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="E48:36"/><b>then </b>
<font color="Maroon">x8</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r8</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E50:36"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">x8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r8</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r8</font> &amp; <font color="Maroon">r8</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<div><i><font color="Green">E136</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_5"/>
<font color="Maroon">r8</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:36_5"/><b>then </b>
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, , , <a class="txt" href="jgraph_5.html#E1:36"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_5"/>
contradiction
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_6"/>
<font color="Maroon">r8</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:36_6"/><b>then </b>
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, , , </i>;<br/><b>hence </b><a NAME="E3:36_6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div>
<a NAME="E53:36"/><b>then </b><i><font color="Green">E138</font></i>: 
1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r8</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</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="E55:36"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E56:36"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E57:36"/>
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> holds  <font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <font color="Olive">p</font>
 
;<br/>
<a NAME="E58:36"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">h2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:36"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E59:36"/><i><font color="Green">E142</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <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="E60:36"/><i><font color="Green">E143</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> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E61:36"/><b>then </b><i><font color="Green">E144</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E62:36"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E63:36"/><b>then </b><i><font color="Green">E145</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <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="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E64:36"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">g0</font> = <font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E66:36"/><i><font color="Green">E146</font></i>: 
(  ex <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> ex <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) implies for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_7"/>
 ex <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> ex <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:36_7"/><i><font color="Green">E147</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_7_1"/>
 ex <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:36_7_1"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/> <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:36_7_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E15:36"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:36_7_1"/><i><font color="Green">E150</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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="E8:36_7_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:36_7_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><i><font color="Green">E152</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> = 
<font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E12:36_7_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">v</font> &amp; <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E13:36_7_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">t</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="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then </b><i><font color="Green">E154</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> = 
<font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_7_1_4_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> or <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_1_4_1_1"><b>case </b></a><a NAME="E1:36_7_1_4_1_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">t</font></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E18:36"><i><font color="Green">E80</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">B0</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g0</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B0</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B0</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/><a NAME="E5:36_7_1_4_1_1"/>
<font color="Maroon">g0</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:36_7_1_4_1_1"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E7:36_7_1_4_1_1"/><i><font color="Green">E158</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><a NAME="E8:36_7_1_4_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E9:36_7_1_4_1_1"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:36_7_1_4_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E11:36_7_1_4_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:36_7_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> )
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:36"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E17:36"><i><font color="Green">E79</font></i></a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:36_7_1_4_1_1"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a>, , , <a class="ref" href="topreal5.html#T14">TOPREAL5:14</a></i>;<br/><a NAME="E15:36_7_1_4_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E16:36_7_1_4_1_1"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E17:36_7_1_4_1_1"/><i><font color="Green">E163</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E18:36_7_1_4_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:36_7_1_4_1_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r2</font> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E26:36"><i><font color="Green">E88</font></i></a></i>;<br/><a NAME="E20:36_7_1_4_1_1"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E21:36_7_1_4_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:36_7_1_4_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E23:36_7_1_4_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">q3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E25:36_7_1_4_1_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a></i>;<br/><i><font color="Green">E166</font></i>: <font color="Maroon">q3</font> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>then </b><i><font color="Green">E167</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">0 <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_1_4_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_1_4_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_7_1_4_1_1_3_1"/>
( <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_1_4_1_1_3_1_1"><b>case </b></a><a NAME="E1:36_7_1_4_1_1_3_1_1"/>
<font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E168</font></i>: <font color="Maroon">q3</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="txt" href="jgraph_5.html#E1:36"><i><font color="Green">E30</font></i></a></i>
;<br/><a NAME="E3:36_7_1_4_1_1_3_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:36_7_1_4_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, , <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="36_7_1_4_1_1_3_1_2"><b>case </b></a><a NAME="E1:36_7_1_4_1_1_3_1_2"/>
<font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E169</font></i>: <font color="Maroon">q3</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, </i>
;<br/><a NAME="E3:36_7_1_4_1_1_3_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:36_7_1_4_1_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, , <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="E29:36_7_1_4_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_1_4_1_2"><b>case </b></a><a NAME="E1:36_7_1_4_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E18:36"><i><font color="Green">E80</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">B0</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g0</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B0</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B0</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/><a NAME="E5:36_7_1_4_1_2"/>
<font color="Maroon">g0</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:36_7_1_4_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E7:36_7_1_4_1_2"/><i><font color="Green">E172</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">t</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><a NAME="E8:36_7_1_4_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E9:36_7_1_4_1_2"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:36_7_1_4_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E11:36_7_1_4_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:36_7_1_4_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> )
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:36"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E17:36"><i><font color="Green">E79</font></i></a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:36_7_1_4_1_2"/><i><font color="Green">E174</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a>, , , <a class="ref" href="topreal5.html#T14">TOPREAL5:14</a></i>;<br/><a NAME="E15:36_7_1_4_1_2"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E16:36_7_1_4_1_2"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E17:36_7_1_4_1_2"/><i><font color="Green">E176</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E18:36_7_1_4_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:36_7_1_4_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r2</font> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E26:36"><i><font color="Green">E88</font></i></a></i>;<br/><a NAME="E20:36_7_1_4_1_2"/><b>then </b><i><font color="Green">E177</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E21:36_7_1_4_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:36_7_1_4_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E23:36_7_1_4_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">q3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E25:36_7_1_4_1_2"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a></i>;<br/><i><font color="Green">E179</font></i>: <font color="Maroon">q3</font> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>then </b><i><font color="Green">E180</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">0 <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_1_4_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_1_4_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_7_1_4_1_2_3_1"/>
( <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_1_4_1_2_3_1_1"><b>case </b></a><a NAME="E1:36_7_1_4_1_2_3_1_1"/>
<font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E182</font></i>: <font color="Maroon">q3</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="txt" href="jgraph_5.html#E1:36"><i><font color="Green">E30</font></i></a></i>
;<br/><a NAME="E3:36_7_1_4_1_2_3_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:36_7_1_4_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, , <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="36_7_1_4_1_2_3_1_2"><b>case </b></a><a NAME="E1:36_7_1_4_1_2_3_1_2"/>
<font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E183</font></i>: <font color="Maroon">q3</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, </i>
;<br/><a NAME="E3:36_7_1_4_1_2_3_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:36_7_1_4_1_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, , <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="E29:36_7_1_4_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_7_1_4_1_3"><b>case </b></a><a NAME="E1:36_7_1_4_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_7_1_4_1_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:36"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E17:36"><i><font color="Green">E79</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:36_7_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_7"/>
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">R</font> =  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>consider </b><font color="Maroon">f2</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> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E69:36"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">f2</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E70:36"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E71:36"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E72:36"/>
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> holds  <font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="funct_2.html#K3">.</a> <font color="Olive">p</font>
 
;<br/>
<a NAME="E73:36"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">h2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:36"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E74:36"/><i><font color="Green">E189</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</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="E75:36"/><i><font color="Green">E190</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> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E76:36"/><b>then </b><i><font color="Green">E191</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E77:36"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E78:36"/><b>then </b><i><font color="Green">E192</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <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="txt" href="jgraph_5.html#E18:36"><i><font color="Green">E80</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E79:36"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</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>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E81:36"/><i><font color="Green">E193</font></i>: 
(  ex <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> ex <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) implies for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_8"/>
 ex <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> ex <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:36_8"/><i><font color="Green">E194</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_8_1"/>
 ex <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:36_8_1"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/> <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:36_8_1"/><i><font color="Green">E196</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:36_8_1"/><i><font color="Green">E197</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</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="E8:36_8_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:36_8_1"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><i><font color="Green">E199</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> = 
<font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E12:36_8_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">v</font> &amp; <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E13:36_8_1"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">t</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="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then </b><i><font color="Green">E201</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> = 
<font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_8_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_8_1_4_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> or <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_8_1_4_1_1"><b>case </b></a><a NAME="E1:36_8_1_4_1_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">t</font></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">B0</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B0</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B0</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/><a NAME="E5:36_8_1_4_1_1"/>
<font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E15:36"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, </i>;<br/><a NAME="E6:36_8_1_4_1_1"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E7:36_8_1_4_1_1"/><i><font color="Green">E204</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><a NAME="E8:36_8_1_4_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E9:36_8_1_4_1_1"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:36_8_1_4_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:36_8_1_4_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:36_8_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E26:36"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:36_8_1_4_1_1"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , , <a class="ref" href="topreal5.html#T14">TOPREAL5:14</a></i>;<br/><a NAME="E15:36_8_1_4_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E16:36_8_1_4_1_1"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E17:36_8_1_4_1_1"/><i><font color="Green">E208</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E18:36_8_1_4_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:36_8_1_4_1_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r2</font> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E20:36_8_1_4_1_1"/><b>then </b><i><font color="Green">E209</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E21:36_8_1_4_1_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:36_8_1_4_1_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E17:36"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E23:36_8_1_4_1_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">q3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E25:36_8_1_4_1_1"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a></i>;<br/><i><font color="Green">E211</font></i>: <font color="Maroon">q3</font> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E34:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E33:36"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
0
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>then </b><i><font color="Green">E212</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">0 <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E34:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8_1_4_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_8_1_4_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_8_1_4_1_1_3_1"/>
( <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E36:36"><i><font color="Green">E101</font></i></a>, <a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_8_1_4_1_1_3_1_1"><b>case </b></a><a NAME="E1:36_8_1_4_1_1_3_1_1"/>
<font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E213</font></i>: <font color="Maroon">q3</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E35:36"><i><font color="Green">E100</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="txt" href="jgraph_5.html#E1:36"><i><font color="Green">E30</font></i></a></i>
;<br/><a NAME="E3:36_8_1_4_1_1_3_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:36_8_1_4_1_1_3_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="txt" href="jgraph_5.html#E33:36"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_5.html#E34:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_5.html#E38:36"><i><font color="Green">E102</font></i></a>, <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="36_8_1_4_1_1_3_1_2"><b>case </b></a><a NAME="E1:36_8_1_4_1_1_3_1_2"/>
<font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E214</font></i>: <font color="Maroon">q3</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E35:36"><i><font color="Green">E100</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, </i>
;<br/><a NAME="E3:36_8_1_4_1_1_3_1_2"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:36_8_1_4_1_1_3_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="txt" href="jgraph_5.html#E33:36"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_5.html#E34:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_5.html#E38:36"><i><font color="Green">E102</font></i></a>, <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="E29:36_8_1_4_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_8_1_4_1_2"><b>case </b></a><a NAME="E1:36_8_1_4_1_2"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">B0</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B0</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B0</font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/><a NAME="E5:36_8_1_4_1_2"/>
<font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E15:36"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E16:36"><i><font color="Green">E77</font></i></a>, </i>;<br/><a NAME="E6:36_8_1_4_1_2"/><b>then </b><i><font color="Green">E216</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E7:36_8_1_4_1_2"/><i><font color="Green">E217</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">t</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><a NAME="E8:36_8_1_4_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E9:36_8_1_4_1_2"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:36_8_1_4_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:36_8_1_4_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:36_8_1_4_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E26:36"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:36_8_1_4_1_2"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , , <a class="ref" href="topreal5.html#T14">TOPREAL5:14</a></i>;<br/><a NAME="E15:36_8_1_4_1_2"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r4</font> where <font color="Olive">r4</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r4</font> &amp; <font color="Olive">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E16:36_8_1_4_1_2"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E17:36_8_1_4_1_2"/><i><font color="Green">E221</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E18:36_8_1_4_1_2"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:36_8_1_4_1_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r2</font> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E20:36_8_1_4_1_2"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E21:36_8_1_4_1_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:36_8_1_4_1_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E17:36"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E23:36_8_1_4_1_2"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">q3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E25:36_8_1_4_1_2"/><i><font color="Green">E223</font></i>: 
( <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a></i>;<br/><i><font color="Green">E224</font></i>: <font color="Maroon">q3</font> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E34:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E33:36"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
0
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>then </b><i><font color="Green">E225</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">0 <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E34:36"><i><font color="Green">E99</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8_1_4_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_8_1_4_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_8_1_4_1_2_3_1"/>
( <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E36:36"><i><font color="Green">E101</font></i></a>, <a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_8_1_4_1_2_3_1_1"><b>case </b></a><a NAME="E1:36_8_1_4_1_2_3_1_1"/>
<font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E226</font></i>: <font color="Maroon">q3</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E35:36"><i><font color="Green">E100</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, <a class="txt" href="jgraph_5.html#E1:36"><i><font color="Green">E30</font></i></a></i>
;<br/><a NAME="E3:36_8_1_4_1_2_3_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:36_8_1_4_1_2_3_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="txt" href="jgraph_5.html#E33:36"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_5.html#E34:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_5.html#E38:36"><i><font color="Green">E102</font></i></a>, <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="36_8_1_4_1_2_3_1_2"><b>case </b></a><a NAME="E1:36_8_1_4_1_2_3_1_2"/>
<font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E227</font></i>: <font color="Maroon">q3</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E35:36"><i><font color="Green">E100</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, </i>
;<br/><a NAME="E3:36_8_1_4_1_2_3_1_2"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:36_8_1_4_1_2_3_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="txt" href="jgraph_5.html#E33:36"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jgraph_5.html#E34:36"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jgraph_5.html#E38:36"><i><font color="Green">E102</font></i></a>, <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="E29:36_8_1_4_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_8_1_4_1_3"><b>case </b></a><a NAME="E1:36_8_1_4_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_8_1_4_1_3"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E26:36"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:36_8_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_8"/>
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> holds <br/><font color="Olive">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E82:36"/><i><font color="Green">E228</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_9"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q3</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:36_9"/>
<font color="Maroon">q3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, , , , , , , , , <a class="txt" href="jgraph_5.html#E14:36"><i><font color="Green">E73</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:36_9"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E83:36"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:36_10"/><i><font color="Green">E230</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_10_1_1"/>
( <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_10_1_1_1"><b>case </b></a><a NAME="E1:36_10_1_1_1"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">p8</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:36_10_1_1_1"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p8</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E4:36_10_1_1_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>1 = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
;<br/><a NAME="E6:36_10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="complex1.html#T161">COMPLEX1:161</a>, <a class="ref" href="square_1.html#T59">SQUARE_1:59</a></i>;<br/><a NAME="E7:36_10_1_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i>, , <a class="ref" href="square_1.html#T110">SQUARE_1:110</a></i>;<br/><b>hence </b><a NAME="E8:36_10_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, , <a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , , <a class="txt" href="jgraph_5.html#E1:36"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_10_1_1_2"><b>case </b></a><a NAME="E1:36_10_1_1_2"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_10_1_1_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:36_10_1_1_2_1"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/> <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><b>then consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_10_1_1_2_1"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E24:36"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:36_10_1_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</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="E7:36_10_1_1_2_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">t2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:36_10_1_1_2_1"/><i><font color="Green">E237</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><div><i><font color="Green">E238</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_10_1_1_2_1_2"/>
<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:36_10_1_1_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, , , <a class="txt" href="jgraph_5.html#E26:36"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_5.html#E1:36"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_10_1_1_2_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10_1_1_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_10_1_1_2_1_3"/>
<font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:36_10_1_1_2_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</a>, , , <a class="txt" href="jgraph_5.html#E26:36"><i><font color="Green">E88</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:36_10_1_1_2_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E12:36_10_1_1_2_1"/><b>then </b><i><font color="Green">E239</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E26:36"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jgraph_5.html#E27:36"><i><font color="Green">E89</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E13:36_10_1_1_2_1"/><i><font color="Green">E240</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E14:36_10_1_1_2_1"/><i><font color="Green">E241</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>then </b><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/><a NAME="E16:36_10_1_1_2_1"/><b>then </b><i><font color="Green">E242</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10_1_1_2_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_10_1_1_2_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_10_1_1_2_1_5_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> )
 </span><b>by </b><i><a class="ref" href="jgraph_5.html#T4" target="_self">Th4</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="36_10_1_1_2_1_5_1_1"><b>case </b></a><a NAME="E1:36_10_1_1_2_1_5_1_1"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_10_1_1_2_1_5_1_1"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jgraph_5.html#E14:36"><i><font color="Green">E73</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_10_1_1_2_1_5_1_2"><b>case </b></a><a NAME="E1:36_10_1_1_2_1_5_1_2"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_10_1_1_2_1_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E21:36"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jgraph_5.html#E30:36"><i><font color="Green">E90</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:36_10_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:36_10_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E84:36"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
