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

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:56"/><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">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  ) )
 ;<br/>

<b>then consider </b><font color="Maroon">p3</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="E3:56"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>
<b>consider </b><font color="Maroon">p3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:56"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p3</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#T8" target="_self">Th8</a></i>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1"/>
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E2:56_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/><a NAME="E3:56_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E4:56_1"/>
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</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">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/><a NAME="E5:56_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</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> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E6:56_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E7:56_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</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="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><a NAME="E8:56_1"/><i><font color="Green">E47</font></i>: 
1 <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</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">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="jgraph_3.html#T10">JGRAPH_3:10</a></i>;<br/><a NAME="E9:56_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</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> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E10:56_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E11:56_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</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="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/><a NAME="E12:56_1"/>
<span class="p1">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E13:56_1"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p2</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="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/><b>hence </b><a NAME="E14:56_1"/>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:56"/><i><font color="Green">E73</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_5.html#T8" target="_self">Th8</a>, <a class="ref" href="jgraph_3.html#T36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E8:56"/><b>then </b><i><font color="Green">E75</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/>
<b>set </b><font color="Maroon">P4</font> =  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>;<br/>
<b>set </b><font color="Maroon">P4b</font> =  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>;<br/>
<a NAME="E9:56"/><i><font color="Green">E77</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="E10:56"/><i><font color="Green">E78</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, <a class="txt" href="jgraph_5.html#E6:56"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E11:56"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E12:56"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E13:56"/><i><font color="Green">E82</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <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#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E14:56"/><i><font color="Green">E83</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <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#T8" target="_self">Th8</a>, <a class="txt" href="jgraph_5.html#E1:56"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E15:56"/>
for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span><br/> for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#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">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> &amp; <font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/><font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:56_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<b>then </b><i><font color="Green">E85</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">g0</font> =  <a href="pscomp_1.html#K13">proj1</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>set </b><font color="Maroon">K0</font> =  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>;<br/>
<b>reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">g0</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></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/>
<b>reconsider </b><font color="Maroon">g3</font> = <font color="Maroon">g2</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E6:56_2"/>
 <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="E7:56_2"/><b>then </b><i><font color="Green">E86</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#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:56_2"/>
 <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="E9:56_2"/><b>then </b><i><font color="Green">E88</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#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:56_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:56_2"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g3</font> <a href="hidden.html#R1">=</a>  <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> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E12:56_2"/><i><font color="Green">E90</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g3</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E13:56_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">g3</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E14:56_2"/>
( not  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a> )
 
<b>by </b><i>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<a NAME="E15:56_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>
<a NAME="E16:56_2"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>;<br/>
<a NAME="E17:56_2"/><b>then </b>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1 <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T38">PCOMPS_1:38</a></i>;<br/>
<a NAME="E18:56_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">g3</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , , <a class="txt" href="jgraph_5.html#E7:56"><i><font color="Green">E73</font></i></a>, <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">g3</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E20:56_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">h</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<a NAME="E21:56_2"/><i><font color="Green">E99</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E22:56_2"/><i><font color="Green">E100</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E23:56_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E24:56_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<i><font color="Green">E103</font></i>:  <a href="real_1.html#K1">-</a> 1 = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<i><font color="Green">E105</font></i>: 1 = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">g0</font> <a href="funct_2.html#K3">.</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="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, , <a class="txt" href="jgraph_5.html#E9:56"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<i><font color="Green">E106</font></i>: <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  = 
<font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, <a class="txt" href="jgraph_5.html#E10:56"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  = 
<font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">g3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
<b>by </b><i>, <a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, <a class="txt" href="jgraph_5.html#E11:56"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E29:56_2"/>
<font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font>
 <b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, , <a class="txt" href="jgraph_5.html#E8:56"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jgraph_5.html#E10:56"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jgraph_5.html#E11:56"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jgraph_5.html#E12:56"><i><font color="Green">E80</font></i></a>, , <a class="txt" href="jgraph_5.html#E13:56"><i><font color="Green">E82</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:56"/><b>then </b>
( <font color="Maroon">p1</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">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <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="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<b>hence </b><a NAME="E17:56"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>


</div>
