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

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>

<b>set </b><font color="Maroon">m</font> =  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">j</font> =  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font>;<br/>
<b>assume </b><a NAME="E1:150"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="jordan24.html#R1">realize-max-dist-in</a> <font color="Maroon">C</font>
 ;<br/>

<b>let </b><font color="Maroon">Jc</font> be  <a href="compts_1.html#V6">compact</a> <a href="jordan21.html#V1">with_the_max_arc</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>, <font color="Maroon">Jd</font> be  <a href="compts_1.html#V6">compact</a> <a href="jordan21.html#V1">with_the_max_arc</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><b>that </b><br/><a NAME="E2:150"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Jc</font> <a href="topreal1.html#R1">is_an_arc_of</a> <span class="p1"><a 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>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E3:150"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">Jd</font> <a href="topreal1.html#R1">is_an_arc_of</a> <span class="p1"><a 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>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E4:150"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Jc</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Jd</font>
 <b>and </b><br/><a NAME="E5:150"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">Jc</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E6:150"/><i><font color="Green">E80</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jc</font>
 <b>and </b><br/><a NAME="E7:150"/><i><font color="Green">E81</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jd</font>
 <b>and </b><br/><a NAME="E8:150"/><i><font color="Green">E82</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Jc</font>
 <b>and </b><br/><a NAME="E9:150"/><i><font color="Green">E83</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Jc</font>
 ;<br/>

<b>set </b><font color="Maroon">l</font> =  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font>;<br/>
<b>set </b><font color="Maroon">LJ</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font>;<br/>
<b>set </b><font color="Maroon">k</font> =  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<b>let </b><font color="Maroon">Ux</font> be   <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="E10:150"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">Ux</font> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E11:150"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:150"/><i><font color="Green">E115</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:150"/><i><font color="Green">E117</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="txt" href="jordan.html#E53"><i><font color="Green">Lemma151</font></i></a></i>;<br/>
<a NAME="E14:150"/><i><font color="Green">E118</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="jordan24.html#D1">JORDAN24:def 1</a></i>;<br/>
<a NAME="E15:150"/><i><font color="Green">E119</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T43">JORDAN21:43</a></i>;<br/>
<a NAME="E16:150"/><i><font color="Green">E120</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jc</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T44">JORDAN21:44</a></i>;<br/>
<a NAME="E17:150"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">Jd</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E18:150"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">Jc</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E19:150"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E20:150"/><i><font color="Green">E177</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="txt" href="jordan.html#E30:150"><i><font color="Green">E187</font></i></a>, , </i>;<br/>
<a NAME="E21:150"/><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="jordan.html#E22:150"><i><font color="Green">E180</font></i></a>, <a class="txt" href="jordan.html#E23:150"><i><font color="Green">E181</font></i></a>, <a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E22:150"/><i><font color="Green">E180</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="jordan.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E23:150"/><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E24:150"/><i><font color="Green">E182</font></i>: 
(  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a 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> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a 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> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan.html#E22:150"><i><font color="Green">E180</font></i></a>, <a class="txt" href="jordan.html#E23:150"><i><font color="Green">E181</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then consider </b><font color="Maroon">Pml</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font><b> such that </b><br/><a NAME="E26:150"/><i><font color="Green">E183</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Jc</font>
 <b>and </b><br/><a NAME="E27:150"/><i><font color="Green">E184</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="jordan.html#T57" target="_self">Th57</a>, , <a class="txt" href="jordan.html#E21:150"><i><font color="Green">E178</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">ml</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font>;<br/>
<a NAME="E28:150"/><i><font color="Green">E185</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E29:150"/><i><font color="Green">E186</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, </i>;<br/>
<a NAME="E30:150"/><i><font color="Green">E187</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E31:150"/><i><font color="Green">E189</font></i>: 
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, </i>;<br/>
<a NAME="E32:150"/><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E33:150"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<a NAME="E34:150"/><b>then </b><i><font color="Green">E192</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E35:150"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan21.html#T42">JORDAN21:42</a></i>;<br/>
<a NAME="E36:150"/><b>then </b>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#D2" target="_self">Def2</a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<a NAME="E37:150"/><b>then </b><i><font color="Green">E193</font></i>: 
not <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E38:150"/><i><font color="Green">E194</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T20">COMPTS_1:20</a></i>;<br/>
<a NAME="E39:150"/><i><font color="Green">E195</font></i>: 
for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="sppol_1.html#V2">vertical</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> holds  <font color="Olive">A</font> is <a href="jordan21.html#V1">with_the_max_arc</a>
 
;<br/>
<a NAME="E40:150"/><i><font color="Green">E196</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="ref" href="jordan.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E41:150"/><b>then </b><i><font color="Green">E198</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font> is <a href="jordan21.html#V1">with_the_max_arc</a>
 
<b>by </b><i><a class="ref" href="jordan.html#D6" target="_self">Def6</a>, <a class="ref" href="jordan.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E42:150"/><i><font color="Green">E199</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font>
 
<b>by </b><i>, <a class="ref" href="jordan.html#D5" target="_self">Def5</a>, <a class="ref" href="jordan21.html#T43">JORDAN21:43</a></i>;<br/>
<a NAME="E43:150"/><b>then </b><i><font color="Green">E252</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E44:150"/><i><font color="Green">E253</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Jd</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E45:150"/><b>then </b><i><font color="Green">E254</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a></i>;<br/>
<a NAME="E46:150"/><i><font color="Green">E257</font></i>: 
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E47:150"/><i><font color="Green">E258</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="jordan.html#D1" target="_self">Def1</a>, , <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E48:150"/>
(  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a 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> &amp;  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a 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> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="jordan.html#T60" target="_self">Th60</a>, <a class="ref" href="jordan.html#D2" target="_self">Def2</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>then consider </b><font color="Maroon">Pkj</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span>, <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font><b> such that </b><br/><a NAME="E50:150"/><i><font color="Green">E259</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Jd</font>
 <b>and </b><br/><a NAME="E51:150"/><i><font color="Green">E260</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, , , </i>;<br/>
<b>set </b><font color="Maroon">kj</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font>;<br/>
<a NAME="E52:150"/><i><font color="Green">E261</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E53:150"/><i><font color="Green">E262</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard7.html#T7">GOBOARD7:7</a></i>;<br/>
<a NAME="E54:150"/><i><font color="Green">E263</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="connsp_1.html#R3">is_a_component_of</a> <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"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T43">CONNSP_1:43</a></i>;<br/>
<a NAME="E55:150"/><i><font color="Green">E264</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E56:150"/><i><font color="Green">E265</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E57:150"/><i><font color="Green">E271</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E58:150"/><i><font color="Green">E272</font></i>: 
(  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</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"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp;  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></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"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <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/>
<a NAME="E59:150"/><i><font color="Green">E273</font></i>: 
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></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"><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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <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/>
<a NAME="E60:150"/>
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, </i>;<br/>
<a NAME="E61:150"/><b>then </b><i><font color="Green">E274</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan.html#D7" target="_self">Def7</a>, , <a class="ref" href="jordan.html#T64" target="_self">Th64</a>, <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E62:150"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> )
 
<b>by </b><i><a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E63:150"/><b>then </b>
( not  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span> &amp; not  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E64:150"/><b>then </b><i><font color="Green">E275</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font>
 
<b>by </b><i>, , <a class="txt" href="jordan.html#E20:150"><i><font color="Green">E177</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E65:150"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<a NAME="E66:150"/><b>then </b><i><font color="Green">E276</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E67:150"/>
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<a NAME="E68:150"/><b>then </b>
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E69:150"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan21.html#T42">JORDAN21:42</a></i>;<br/>
<a NAME="E70:150"/><b>then </b>
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E71:150"/><b>then </b><i><font color="Green">E277</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan.html#T63" target="_self">Th63</a>, <a class="ref" href="jordan.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan.html#T60" target="_self">Th60</a>, , <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/>
<a NAME="E72:150"/><i><font color="Green">E278</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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><b>that </b><br/><a NAME="E1:150_1"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E2:150_1"/><i><font color="Green">E280</font></i>: 
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E3:150_1"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:150_1"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<a NAME="E6:150_1"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , <a class="ref" href="jordan.html#T63" target="_self">Th63</a>, <a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E7:150_1"/><b>then </b><i><font color="Green">E284</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:150_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jc</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jd</font> )
 </span><b>by </b><i>, , <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="150_1_1_1"><b>suppose </b></a><a NAME="E1:150_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jc</font>
 ;<br/><div class="add"><a NAME="E2:150_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jc</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:150_1_1_1"/><b>then </b><i><font color="Green">E285</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E22:150"><i><font color="Green">E180</font></i></a>, <a class="txt" href="jordan.html#E23:150"><i><font color="Green">E181</font></i></a>, <a class="ref" href="jordan21.html#T42">JORDAN21:42</a></i>;<br/><a NAME="E4:150_1_1_1"/>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="ref" href="jordan.html#D7" target="_self">Def7</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/><a NAME="E5:150_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:150_1_1_1"/><b>then </b>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/><a NAME="E7:150_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E8:150_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_1_1_2"><b>suppose </b></a><a NAME="E1:150_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jd</font>
 ;<br/><div class="add"><a NAME="E2:150_1_1_2"/><b>then </b><i><font color="Green">E286</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:150_1_1_2"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="jordan.html#D1" target="_self">Def1</a>, , <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><a NAME="E4:150_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E5:150_1_1_2"/><b>then </b><i><font color="Green">E288</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> &amp;  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T130">PSCOMP_1:130</a>, <a class="ref" href="pscomp_1.html#T132">PSCOMP_1:132</a></i>;<br/><a NAME="E6:150_1_1_2"/><i><font color="Green">E289</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#D5" target="_self">Def5</a>, <a class="ref" href="jordan.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan.html#T58" target="_self">Th58</a>, <a class="ref" href="sprect_1.html#T17">SPRECT_1:17</a></i>;<br/><a NAME="E7:150_1_1_2"/><i><font color="Green">E290</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  &amp;  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="sprect_1.html#T62">SPRECT_1:62</a>, <a class="ref" href="sprect_1.html#T65">SPRECT_1:65</a></i>;<br/><a NAME="E8:150_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> &amp;  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E9:150_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/><a NAME="E10:150_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:150_1_1_2"/><b>then </b><i><font color="Green">E291</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan.html#D5" target="_self">Def5</a>, <a class="ref" href="jordan21.html#T41">JORDAN21:41</a></i>;<br/><a NAME="E12:150_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="ref" href="jordan.html#D7" target="_self">Def7</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/><a NAME="E13:150_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E14:150_1_1_2"/><b>then </b>
 <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="jordan.html#T60" target="_self">Th60</a>, , <a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/><a NAME="E15:150_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E16:150_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> as   <a href="struct_0.html#NM2">Subset</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"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:150_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:150_2"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> or <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:150_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E75:150"/><b>then </b><i><font color="Green">E292</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E76:150"/><b>then </b>
 <a href="connsp_3.html#K6">Component_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="connsp_3.html#T27">CONNSP_3:27</a></i>;<br/>
<a NAME="E77:150"/><b>then </b><i><font color="Green">E293</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="connsp_3.html#T26">CONNSP_3:26</a></i>;<br/>
<a NAME="E78:150"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Ux</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E79:150"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T52">JORDAN1:52</a></i>;<br/>
<a NAME="E80:150"/><b>then </b>
<font color="Maroon">X</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E81:150"/><b>then </b><i><font color="Green">E295</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, <a class="ref" href="jordan.html#T62" target="_self">Th62</a>, , <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/>
<a NAME="E82:150"/><i><font color="Green">E296</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, , <a class="ref" href="jordan.html#T59" target="_self">Th59</a>, <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E83:150"/><i><font color="Green">E297</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="jordan1.html#T1">JORDAN1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">AR</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>, <font color="Maroon">BR</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">CR</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">DR</font> = <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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E29:150"><i><font color="Green">E186</font></i></a>, <a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, <a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan1.html#T1">JORDAN1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">Pcm</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>, <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font>, <font color="Maroon">fcm</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="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E86:150"/><i><font color="Green">E298</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fcm</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">Pcm</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fcm</font> )
 <b>by </b><i/>;<br/>
<a NAME="E87:150"/><i><font color="Green">E299</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E30:150"><i><font color="Green">E187</font></i></a>, <a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, <a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E88:150"/><i><font color="Green">E300</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E42:150"><i><font color="Green">E199</font></i></a>, <a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>thus </b><a NAME="E89:150"/>
<font color="Maroon">Ux</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:150_3"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">Ux</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan.html#T62" target="_self">Th62</a>, <a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan2c.html#D3">JORDAN2C:def 3</a><br/>

<b>assume </b><a NAME="E2:150_3"/>
not <font color="Maroon">Ux</font> is <a href="jordan2c.html#V1">Bounded</a>
 ;<br/>

<a NAME="E3:150_3"/><b>then </b>
not <font color="Maroon">Ux</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,10
 
<b>by </b><i><a class="ref" href="jordan2c.html#T16">JORDAN2C:16</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:150_3"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ux</font>
 <b>and </b><br/><a NAME="E6:150_3"/><i><font color="Green">E303</font></i>: 
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,10
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:150_3"/><i><font color="Green">E304</font></i>: 
 <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3 <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,10
 
<b>by </b><i>, , <a class="ref" href="jordan.html#T50" target="_self">Th50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">xs</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span>, <font color="Maroon">us</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <span class="p3">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, , , <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E10:150_3"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">Ux</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="sprect_3.html#T18">SPRECT_3:18</a></i>;<br/>
<a NAME="E11:150_3"/>
 <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="jordan.html#T62" target="_self">Th62</a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/>
<a NAME="E12:150_3"/><b>then </b>
<font color="Maroon">Ux</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E13:150_3"/><b>then </b><i><font color="Green">E306</font></i>: 
<font color="Maroon">Ux</font> <a href="topreal4.html#NR3" target="_self">is_Region</a> 
 
<b>by </b><i>, <a class="ref" href="topreal4.html#D3">TOPREAL4:def 3</a></i>;<br/>
<a NAME="E14:150_3"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,10
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">P1</font> being   <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><b> such that </b><br/><a NAME="E16:150_3"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">P1</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span>,<font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E17:150_3"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ux</font>
 <b>by </b><i><a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, , , , , <a class="ref" href="topreal4.html#T30">TOPREAL4:30</a></i>;<br/>
<a NAME="E18:150_3"/><i><font color="Green">E309</font></i>: 
<font color="Maroon">P1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span>,<font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="topreal4.html#T3">TOPREAL4:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">P1</font> as   <a href="struct_0.html#NM2">Subset</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"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i>, <a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E20:150_3"/><i><font color="Green">E310</font></i>: 
<font color="Maroon">P1</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T11">JORDAN6:11</a></i>;<br/>
<a NAME="E21:150_3"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, </i>;<br/>
<a NAME="E22:150_3"/><i><font color="Green">E312</font></i>: 
<font color="Maroon">P2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E23:150_3"/><b>then </b><i><font color="Green">E313</font></i>: 
<font color="Maroon">P2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jc</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E24:150_3"/><i><font color="Green">E314</font></i>: 
<font color="Maroon">P2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<i><font color="Green">E316</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7">TOPREAL3:7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a></i>
;<br/>
<a NAME="E26:150_3"/><b>then </b><i><font color="Green">E317</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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E27:150_3"/><i><font color="Green">E318</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <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/>
<a NAME="E28:150_3"/><i><font color="Green">E319</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E29:150_3"/><i><font color="Green">E320</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E30:150_3"/>
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, , , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, </i>;<br/>
<a NAME="E31:150_3"/><b>then </b><i><font color="Green">E321</font></i>: 
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E47:150"><i><font color="Green">E258</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:150_3"/>
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
 
;<br/>
<a NAME="E33:150_3"/><b>then </b><i><font color="Green">E322</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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan.html#E46:150"><i><font color="Green">E257</font></i></a>, <a class="txt" href="jordan.html#E50:150"><i><font color="Green">E259</font></i></a>, <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/>
<a NAME="E34:150_3"/><i><font color="Green">E323</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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jc</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:150_3_2"/>
not  <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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jc</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:150_3_2"/><i><font color="Green">E324</font></i>: 
<font color="Maroon">q</font> <a href="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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:150_3_2"/><i><font color="Green">E325</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E53:150"><i><font color="Green">E262</font></i></a></i>;<br/>
<a NAME="E6:150_3_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan.html#T64" target="_self">Th64</a>, <a class="txt" href="jordan.html#E53:150"><i><font color="Green">E262</font></i></a>, , <a class="txt" href="jordan.html#E50:150"><i><font color="Green">E259</font></i></a>, <a class="txt" href="jordan.html#E45:150"><i><font color="Green">E254</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E7:150_3_2"/><b>then </b><i><font color="Green">E326</font></i>: 
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E46:150"><i><font color="Green">E257</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:150_3_2"/>
<font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="jordan.html#E51:150"><i><font color="Green">E260</font></i></a>, <a class="txt" href="jordan.html#E53:150"><i><font color="Green">E262</font></i></a>, <a class="ref" href="jordan.html#D1" target="_self">Def1</a>, , <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E9:150_3_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<a NAME="E10:150_3_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jc</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E54:150"><i><font color="Green">E263</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:150_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E55:150"><i><font color="Green">E264</font></i></a>, <a class="txt" href="jordan.html#E22:150"><i><font color="Green">E180</font></i></a>, <a class="txt" href="jordan.html#E23:150"><i><font color="Green">E181</font></i></a>, <a class="ref" href="jordan21.html#T42">JORDAN21:42</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">n</font> =  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>;<br/>
<a NAME="E35:150_3"/><i><font color="Green">E327</font></i>: 
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, </i>;<br/>
<a NAME="E36:150_3"/><i><font color="Green">E328</font></i>: 
 <a href="tops_1.html#K2">Fr</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i/>;<br/>
<a NAME="E37:150_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E38:150_3"/><b>then </b><i><font color="Green">E329</font></i>: 
<font color="Maroon">P1</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E53:150"><i><font color="Green">E262</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E39:150_3"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E40:150_3"/><b>then </b>
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E41:150_3"/><b>then </b><i><font color="Green">E330</font></i>: 
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, <a class="txt" href="jordan.html#E54:150"><i><font color="Green">E263</font></i></a>, <a class="txt" href="jordan.html#E55:150"><i><font color="Green">E264</font></i></a>, <a class="ref" href="connsp_1.html#T23">CONNSP_1:23</a></i>;<br/>
<a NAME="E42:150_3"/>
<font color="Maroon">P1</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="E43:150_3"/><b>then </b><i><font color="Green">E331</font></i>: 
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> 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="E44:150_3"/><i><font color="Green">E332</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E56:150"><i><font color="Green">E265</font></i></a>, , , <a class="ref" href="jordan5c.html#D1">JORDAN5C:def 1</a></i>;<br/>
<a NAME="E45:150_3"/><b>then </b><i><font color="Green">E333</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E46:150_3"/><i><font color="Green">E335</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</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/>
<b>set </b><font color="Maroon">alpha</font> =  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>;<br/>
<a NAME="E47:150_3"/><i><font color="Green">E336</font></i>: 
 <a href="real_1.html#K1">-</a> 3 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, , </i>;<br/>
<a NAME="E48:150_3"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan21.html#T41">JORDAN21:41</a></i>;<br/>
<a NAME="E49:150_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E46:150"><i><font color="Green">E257</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E50:150_3"/><b>then </b>
not <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, , <a class="txt" href="jordan.html#E45:150"><i><font color="Green">E254</font></i></a>, <a class="txt" href="jordan.html#E52"><i><font color="Green">Lemma149</font></i></a>, <a class="txt" href="jordan.html#E34:150"><i><font color="Green">E192</font></i></a>, <a class="txt" href="jordan.html#E47:150"><i><font color="Green">E258</font></i></a></i>;<br/>
<a NAME="E51:150_3"/><b>then </b><i><font color="Green">E338</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span>, <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="jordan16.html#T39">JORDAN16:39</a></i>;<br/>
<a NAME="E52:150_3"/><i><font color="Green">E339</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="jordan16.html#T5">JORDAN16:5</a></i>;<br/>
<a NAME="E53:150_3"/><b>then </b><i><font color="Green">E342</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jc</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E43:150"><i><font color="Green">E252</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E54:150_3"/><i><font color="Green">E343</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E58:150"><i><font color="Green">E272</font></i></a>, <a class="txt" href="jordan.html#E44:150"><i><font color="Green">E253</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>consider </b><font color="Maroon">Pdx</font> being    <a href="borsuk_2.html#M1">Path</a> of <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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span>, <font color="Maroon">fdx</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="topreal1.html#K3">LSeg</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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E56:150_3"/><i><font color="Green">E344</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fdx</font> <a href="hidden.html#R1">=</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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">Pdx</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fdx</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">PJc</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a 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>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fJc</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">Jc</font></span>)</span><b> such that </b><br/><a NAME="E58:150_3"/><i><font color="Green">E345</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fJc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Jc</font> &amp; <font color="Maroon">PJc</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fJc</font> )
 <b>by </b><i><a class="ref" href="jordan.html#T57" target="_self">Th57</a>, </i>;<br/>
<b>consider </b><font color="Maroon">PJd</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a 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>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fJd</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">Jd</font></span>)</span><b> such that </b><br/><a NAME="E60:150_3"/><i><font color="Green">E348</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fJd</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Jd</font> &amp; <font color="Maroon">PJd</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fJd</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">Palpha</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span>, <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>, <font color="Maroon">falpha</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#K5">Segment</a> <font color="Maroon">P1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E62:150_3"/><i><font color="Green">E349</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">falpha</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> &amp; <font color="Maroon">Palpha</font> <a href="hidden.html#R1">=</a> <font color="Maroon">falpha</font> )
 <b>by </b><i><a class="txt" href="jordan.html#E57:150"><i><font color="Green">E271</font></i></a>, </i>;<br/>
<a NAME="E63:150_3"/>
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E64:150_3"/><b>then </b><i><font color="Green">E352</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> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 
;<br/>
<a NAME="E65:150_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJc</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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, <a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="txt" href="jordan.html#E66:150"><i><font color="Green">E276</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">PJc</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">AR</font>,<font color="Maroon">BR</font> <b>by </b><i/>;<br/>
<a NAME="E67:150_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJd</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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, <a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, <a class="txt" href="jordan.html#E71:150"><i><font color="Green">E277</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">PJd</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">AR</font>,<font color="Maroon">BR</font> <b>by </b><i/>;<br/>
<a NAME="E69:150_3"/><i><font color="Green">E353</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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="jordan.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , , <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E70:150_3"/><i><font color="Green">E354</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, , , <a class="txt" href="jordan.html#E53:150"><i><font color="Green">E262</font></i></a>, </i>;<br/>
<a NAME="E71:150_3"/><i><font color="Green">E355</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E72:150_3"/><i><font color="Green">E356</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E73:150_3"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="jordan.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E74:150_3"/><b>then </b><i><font color="Green">E360</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jc</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E75:150_3"/><i><font color="Green">E361</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E76:150_3"/><i><font color="Green">E362</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E77:150_3"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="jordan.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E78:150_3"/><b>then </b><i><font color="Green">E363</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jc</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>consider </b><font color="Maroon">Plx</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span>, <font color="Maroon">flx</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E80:150_3"/><i><font color="Green">E525</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">flx</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">Plx</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">flx</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">PCX</font> = <span class="p1">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font>;<br/>
<a NAME="E81:150_3"/><i><font color="Green">E527</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pcm</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Plx</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E82:150_3"/><i><font color="Green">E528</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:150_3_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Jd</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:150_3_3"/><i><font color="Green">E529</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font>
 <b>and </b><br/><a NAME="E4:150_3_3"/><i><font color="Green">E530</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jd</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:150_3_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, , <a class="txt" href="jordan.html#E75:150"><i><font color="Green">E292</font></i></a>, <a class="txt" href="jordan.html#E20:150"><i><font color="Green">E177</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:150_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E41:150"><i><font color="Green">E198</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E83:150_3"/><i><font color="Green">E531</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="jordan.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E84:150_3"/><i><font color="Green">E532</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:150_3_4"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a> <font color="Maroon">Jd</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:150_3_4"/><i><font color="Green">E533</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span>
 <b>and </b><br/><a NAME="E4:150_3_4"/><i><font color="Green">E534</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jd</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:150_3_4"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E78:150"><i><font color="Green">E294</font></i></a>, <a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, , <a class="txt" href="jordan.html#E77:150"><i><font color="Green">E293</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:150_3_4"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:150_3_4"/><b>then </b>
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E78:150"><i><font color="Green">E294</font></i></a>, <a class="txt" href="jordan.html#E21:150"><i><font color="Green">E178</font></i></a>, <a class="txt" href="jordan.html#E20:150"><i><font color="Green">E177</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:150_3_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E40:150"><i><font color="Green">E196</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E85:150_3"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E45:150"><i><font color="Green">E254</font></i></a>, <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E86:150_3"/><b>then </b><i><font color="Green">E535</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan.html#T63" target="_self">Th63</a>, <a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="txt" href="jordan.html#E45:150"><i><font color="Green">E254</font></i></a>, <a class="txt" href="jordan.html#E47:150"><i><font color="Green">E258</font></i></a>, <a class="txt" href="jordan.html#E46:150"><i><font color="Green">E257</font></i></a>, <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/>
<a NAME="E87:150_3"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E88:150_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E89:150_3"/><b>then </b><i><font color="Green">E536</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E90:150_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:150_3_5"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:150_3_5"/><b>then </b><i><font color="Green">E537</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E3:150_3_5"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan.html#D7" target="_self">Def7</a>, <a class="txt" href="jordan.html#E46:150"><i><font color="Green">E257</font></i></a>, <a class="txt" href="jordan.html#E47:150"><i><font color="Green">E258</font></i></a>, <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, <a class="txt" href="jordan.html#E45:150"><i><font color="Green">E254</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E4:150_3_5"/><b>then </b>
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E81:150"><i><font color="Green">E295</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:150_3_5"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E81:150"><i><font color="Green">E295</font></i></a>, <a class="txt" href="jordan.html#E77:150"><i><font color="Green">E293</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E91:150_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E92:150_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E93:150_3"/><b>then </b><i><font color="Green">E538</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E94:150_3"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_3_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:150_3_6"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#R2">meets</a> <font color="Maroon">Jd</font>
 ;<br/>

<a NAME="E2:150_3_6"/><b>then </b>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jd</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E3:150_3_6"/><b>then </b>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E20:150"><i><font color="Green">E177</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:150_3_6"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E40:150"><i><font color="Green">E196</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E95:150_3"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E78:150"><i><font color="Green">E294</font></i></a>, <a class="txt" href="jordan.html#E81:150"><i><font color="Green">E295</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>
<a NAME="E96:150_3"/><b>then </b><i><font color="Green">E539</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<b>by </b><i>, , , <a class="txt" href="jordan.html#E75:150"><i><font color="Green">E292</font></i></a>, , <a class="ref" href="xboole_1.html#T114">XBOOLE_1:114</a></i>;<br/>
<a NAME="E97:150_3"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, <a class="txt" href="jordan.html#E77:150"><i><font color="Green">E293</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E98:150_3"/><b>then </b><i><font color="Green">E540</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E99:150_3"/><i><font color="Green">E541</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="jordan.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E100:150_3"/><b>then </b><i><font color="Green">E542</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E101:150_3"/><i><font color="Green">E543</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E102:150_3"/><b>then </b><i><font color="Green">E544</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:150_3_7"/>
( <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> 0 or <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_1"><b>suppose </b></a><a NAME="E1:150_3_7_1"/><i><font color="Green">E545</font></i>: 
<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:150_3_7_1_1"/>
(  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i>, <a class="txt" href="jordan.html#E5:150_3"><i><font color="Green">E302</font></i></a>, <a class="txt" href="jordan.html#E39"><i><font color="Green">Lemma137</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_1_1_1"><b>suppose </b></a><a NAME="E1:150_3_7_1_1_1"/><i><font color="Green">E546</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pnld</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fnld</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:150_3_7_1_1_1"/><i><font color="Green">E549</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fnld</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pnld</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fnld</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Pldd</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fldd</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="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:150_3_7_1_1_1"/><i><font color="Green">E552</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fldd</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pldd</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fldd</font> )
 <b>by </b><i/>;<br/><a NAME="E6:150_3_7_1_1_1"/><i><font color="Green">E553</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="txt" href="jordan.html#E16:150"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><a NAME="E7:150_3_7_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V2">vertical</a>
 <b>by </b><i><a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/><a NAME="E8:150_3_7_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E13:150_3"><i><font color="Green">E306</font></i></a>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, , <a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/><a NAME="E9:150_3_7_1_1_1"/><b>then </b><i><font color="Green">E554</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnld</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pldd</font>;<br/><a NAME="E10:150_3_7_1_1_1"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, </i>;<br/><a NAME="E11:150_3_7_1_1_1"/><b>then </b><i><font color="Green">E556</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 <b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E12:150_3_7_1_1_1"/><i><font color="Green">E557</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnld</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pldd</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Palpha</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pnld</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pldd</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:150_3_7_1_1_1"/><b>then </b><i><font color="Green">E558</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJd</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnld</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pldd</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E71:150"><i><font color="Green">E277</font></i></a>, <a class="txt" href="jordan.html#E61:150"><i><font color="Green">E274</font></i></a>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E17:150_3"><i><font color="Green">E308</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, <a class="txt" href="jordan.html#E82:150"><i><font color="Green">E296</font></i></a>, <a class="txt" href="jordan.html#E87:150"><i><font color="Green">E299</font></i></a>, </i>;<br/><a NAME="E14:150_3_7_1_1_1"/><i><font color="Green">E559</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:150_3_7_1_1_1"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E16:150_3"><i><font color="Green">E307</font></i></a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:150_3_7_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnld</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pldd</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E18:150_3"><i><font color="Green">E309</font></i></a>, , , <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, <a class="txt" href="jordan.html#E83:150"><i><font color="Green">E297</font></i></a>, <a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnld</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pldd</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E19:150_3_7_1_1_1"/><i><font color="Green">E560</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/><a NAME="E20:150_3_7_1_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E21:150_3_7_1_1_1"/><b>then </b>
( <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJd</font> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnld</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pldd</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E22:150_3_7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, <a class="txt" href="jordan.html#E20:150_3"><i><font color="Green">E310</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_1_1_2"><b>suppose </b></a><a NAME="E1:150_3_7_1_1_2"/><i><font color="Green">E561</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pnd</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fnd</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:150_3_7_1_1_2"/><i><font color="Green">E564</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fnd</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pnd</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fnd</font> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font>;<br/><a NAME="E4:150_3_7_1_1_2"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E5:150_3_7_1_1_2"/><b>then </b><i><font color="Green">E565</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan.html#T23" target="_self">Th23</a>, <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="ref" href="sppol_1.html#D2">SPPOL_1:def 2</a></i>;<br/><a NAME="E6:150_3_7_1_1_2"/><b>then </b><i><font color="Green">E566</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V1">horizontal</a>
 <b>by </b><i><a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a></i>;<br/><a NAME="E7:150_3_7_1_1_2"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="jgraph_6.html#T11">JGRAPH_6:11</a></i>;<br/><a NAME="E8:150_3_7_1_1_2"/><b>then </b><i><font color="Green">E567</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jordan.html#T23" target="_self">Th23</a>, <a class="txt" href="jordan.html#E13:150_3"><i><font color="Green">E306</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, <a class="ref" href="goboard7.html#T66">GOBOARD7:66</a></i>;<br/><a NAME="E9:150_3_7_1_1_2"/><b>then </b><i><font color="Green">E568</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:150_3_7_1_1_2"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan.html#E16:150_3"><i><font color="Green">E307</font></i></a>, <a class="txt" href="jordan.html#E86:150"><i><font color="Green">E298</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E11:150_3_7_1_1_2"/><b>then </b><i><font color="Green">E569</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 <b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E12:150_3_7_1_1_2"/><i><font color="Green">E570</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Palpha</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pnd</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:150_3_7_1_1_2"/><b>then </b><i><font color="Green">E571</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jd</font>
 <b>by </b><i><a class="txt" href="jordan.html#E61:150"><i><font color="Green">E274</font></i></a>, <a class="txt" href="jordan.html#E82:150"><i><font color="Green">E296</font></i></a>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E18:150_3"><i><font color="Green">E309</font></i></a>, <a class="ref" href="xboole_1.html#T114">XBOOLE_1:114</a></i>;<br/><a NAME="E14:150_3_7_1_1_2"/><i><font color="Green">E572</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E17:150_3"><i><font color="Green">E308</font></i></a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:150_3_7_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E83:150"><i><font color="Green">E297</font></i></a>, <a class="txt" href="jordan.html#E20:150_3"><i><font color="Green">E310</font></i></a>, , , <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E18:150_3_7_1_1_2"/><i><font color="Green">E573</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/><a NAME="E19:150_3_7_1_1_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E20:150_3_7_1_1_2"/><b>then </b>
( <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJd</font> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E21:150_3_7_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E23:150_3"><i><font color="Green">E313</font></i></a>, <a class="txt" href="jordan.html#E71:150"><i><font color="Green">E277</font></i></a>, <a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_1_1_3"><b>suppose </b></a><a NAME="E1:150_3_7_1_1_3"/><i><font color="Green">E574</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pnd</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fnd</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:150_3_7_1_1_3"/><i><font color="Green">E575</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fnd</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pnd</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fnd</font> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font>;<br/><a NAME="E4:150_3_7_1_1_3"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E5:150_3_7_1_1_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="ref" href="sppol_1.html#D2">SPPOL_1:def 2</a></i>;<br/><a NAME="E6:150_3_7_1_1_3"/><b>then </b><i><font color="Green">E576</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V1">horizontal</a>
 <b>by </b><i><a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, , <a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a></i>;<br/><a NAME="E7:150_3_7_1_1_3"/><i><font color="Green">E577</font></i>: 
<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E8:150_3_7_1_1_3"/>
( <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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, , <a class="ref" href="jgraph_6.html#T11">JGRAPH_6:11</a></i>;<br/><a NAME="E9:150_3_7_1_1_3"/><b>then </b><i><font color="Green">E578</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, <a class="txt" href="jordan.html#E13:150_3"><i><font color="Green">E306</font></i></a>, <a class="ref" href="goboard7.html#T66">GOBOARD7:66</a></i>;<br/><a NAME="E10:150_3_7_1_1_3"/><b>then </b><i><font color="Green">E579</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:150_3_7_1_1_3"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan.html#E16:150_3"><i><font color="Green">E307</font></i></a>, <a class="txt" href="jordan.html#E88:150"><i><font color="Green">E300</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E12:150_3_7_1_1_3"/><b>then </b><i><font color="Green">E580</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 <b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E13:150_3_7_1_1_3"/><i><font color="Green">E581</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Palpha</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pnd</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E14:150_3_7_1_1_3"/><b>then </b><i><font color="Green">E582</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jd</font>
 <b>by </b><i><a class="txt" href="jordan.html#E61:150"><i><font color="Green">E274</font></i></a>, <a class="txt" href="jordan.html#E82:150"><i><font color="Green">E296</font></i></a>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E18:150_3"><i><font color="Green">E309</font></i></a>, <a class="ref" href="xboole_1.html#T114">XBOOLE_1:114</a></i>;<br/><a NAME="E15:150_3_7_1_1_3"/><i><font color="Green">E583</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E17:150_3"><i><font color="Green">E308</font></i></a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:150_3_7_1_1_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E83:150"><i><font color="Green">E297</font></i></a>, <a class="txt" href="jordan.html#E20:150_3"><i><font color="Green">E310</font></i></a>, , , <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E19:150_3_7_1_1_3"/><i><font color="Green">E584</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/><a NAME="E20:150_3_7_1_1_3"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E21:150_3_7_1_1_3"/><b>then </b>
( <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJd</font> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnd</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E22:150_3_7_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E23:150_3"><i><font color="Green">E313</font></i></a>, <a class="txt" href="jordan.html#E71:150"><i><font color="Green">E277</font></i></a>, <a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_1_1_4"><b>suppose </b></a><a NAME="E1:150_3_7_1_1_4"/><i><font color="Green">E585</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pnpd</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fnpd</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:150_3_7_1_1_4"/><i><font color="Green">E588</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fnpd</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pnpd</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fnpd</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Ppdd</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fpdd</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="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:150_3_7_1_1_4"/><i><font color="Green">E591</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fpdd</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Ppdd</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fpdd</font> )
 <b>by </b><i/>;<br/><a NAME="E6:150_3_7_1_1_4"/><i><font color="Green">E592</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="txt" href="jordan.html#E18:150"><i><font color="Green">E122</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><a NAME="E7:150_3_7_1_1_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V2">vertical</a>
 <b>by </b><i><a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/><a NAME="E8:150_3_7_1_1_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E13:150_3"><i><font color="Green">E306</font></i></a>, , , , <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/><a NAME="E9:150_3_7_1_1_4"/><b>then </b><i><font color="Green">E593</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpd</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppdd</font>;<br/><a NAME="E10:150_3_7_1_1_4"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="txt" href="jordan.html#E51"><i><font color="Green">Lemma148</font></i></a></i>;<br/><a NAME="E11:150_3_7_1_1_4"/><b>then </b><i><font color="Green">E594</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jd</font>
 <b>by </b><i><a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E12:150_3_7_1_1_4"/><i><font color="Green">E595</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpd</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppdd</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Palpha</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pnpd</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Ppdd</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:150_3_7_1_1_4"/><b>then </b><i><font color="Green">E596</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJd</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpd</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppdd</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E71:150"><i><font color="Green">E277</font></i></a>, <a class="txt" href="jordan.html#E61:150"><i><font color="Green">E274</font></i></a>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E17:150_3"><i><font color="Green">E308</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, <a class="txt" href="jordan.html#E82:150"><i><font color="Green">E296</font></i></a>, <a class="txt" href="jordan.html#E1:150_3"><i><font color="Green">E301</font></i></a>, </i>;<br/><a NAME="E14:150_3_7_1_1_4"/><i><font color="Green">E597</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:150_3_7_1_1_4"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E16:150_3"><i><font color="Green">E307</font></i></a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:150_3_7_1_1_4"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpd</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppdd</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E18:150_3"><i><font color="Green">E309</font></i></a>, , , <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, <a class="txt" href="jordan.html#E83:150"><i><font color="Green">E297</font></i></a>, <a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpd</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppdd</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E19:150_3_7_1_1_4"/><i><font color="Green">E598</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/><a NAME="E20:150_3_7_1_1_4"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E21:150_3_7_1_1_4"/><b>then </b>
( <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJd</font> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plx</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpd</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppdd</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E22:150_3_7_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, <a class="txt" href="jordan.html#E20:150_3"><i><font color="Green">E310</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_2"><b>suppose </b></a><a NAME="E1:150_3_7_2"/><i><font color="Green">E599</font></i>: 
<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:150_3_7_2_1"/>
(  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> or  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i>, <a class="txt" href="jordan.html#E5:150_3"><i><font color="Green">E302</font></i></a>, <a class="ref" href="jordan.html#T30" target="_self">Th30</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_2_1_1"><b>suppose </b></a><a NAME="E1:150_3_7_2_1_1"/><i><font color="Green">E600</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pnlg</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fnlg</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:150_3_7_2_1_1"/><i><font color="Green">E603</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fnlg</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pnlg</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fnlg</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Plgc</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">flgc</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="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:150_3_7_2_1_1"/><i><font color="Green">E606</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">flgc</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Plgc</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">flgc</font> )
 <b>by </b><i/>;<br/><a NAME="E6:150_3_7_2_1_1"/><i><font color="Green">E607</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="txt" href="jordan.html#E15:150"><i><font color="Green">E119</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><a NAME="E7:150_3_7_2_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V2">vertical</a>
 <b>by </b><i><a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/><a NAME="E8:150_3_7_2_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E13:150_3"><i><font color="Green">E306</font></i></a>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, , <a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/><a NAME="E9:150_3_7_2_1_1"/><b>then </b><i><font color="Green">E608</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnlg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plgc</font>;<br/><a NAME="E10:150_3_7_2_1_1"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="ref" href="jordan.html#T40" target="_self">Th40</a></i>;<br/><a NAME="E11:150_3_7_2_1_1"/><b>then </b><i><font color="Green">E609</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E12:150_3_7_2_1_1"/><i><font color="Green">E610</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnlg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plgc</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pdx</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Palpha</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pnlg</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Plgc</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:150_3_7_2_1_1"/><b>then </b><i><font color="Green">E611</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnlg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plgc</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="txt" href="jordan.html#E52:150"><i><font color="Green">E261</font></i></a>, <a class="txt" href="jordan.html#E59:150"><i><font color="Green">E273</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E17:150_3"><i><font color="Green">E308</font></i></a>, , <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, </i>;<br/><a NAME="E14:150_3_7_2_1_1"/><i><font color="Green">E612</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnlg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plgc</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnlg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plgc</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:150_3_7_2_1_1"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E16:150_3"><i><font color="Green">E307</font></i></a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:150_3_7_2_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnlg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plgc</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan.html#E18:150_3"><i><font color="Green">E309</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, , , , <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">v</font> =  <a href="borsuk_2.html#K2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnlg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plgc</font></span>)</span> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i><a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E19:150_3_7_2_1_1"/><i><font color="Green">E613</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/><a NAME="E20:150_3_7_2_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E21:150_3_7_2_1_1"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJc</font> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnlg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plgc</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E22:150_3_7_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, <a class="txt" href="jordan.html#E66:150"><i><font color="Green">E276</font></i></a>, <a class="txt" href="jordan.html#E20:150_3"><i><font color="Green">E310</font></i></a>, <a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_2_1_2"><b>suppose </b></a><a NAME="E1:150_3_7_2_1_2"/><i><font color="Green">E614</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pnc</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fnc</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:150_3_7_2_1_2"/><i><font color="Green">E617</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fnc</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pnc</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fnc</font> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font>;<br/><a NAME="E4:150_3_7_2_1_2"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E5:150_3_7_2_1_2"/><b>then </b><i><font color="Green">E618</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E12:150"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="ref" href="sppol_1.html#D2">SPPOL_1:def 2</a></i>;<br/><a NAME="E6:150_3_7_2_1_2"/><b>then </b><i><font color="Green">E619</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V1">horizontal</a>
 <b>by </b><i>, <a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a></i>;<br/><a NAME="E7:150_3_7_2_1_2"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, , <a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="ref" href="jgraph_6.html#T11">JGRAPH_6:11</a></i>;<br/><a NAME="E8:150_3_7_2_1_2"/><b>then </b><i><font color="Green">E620</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E12:150"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan.html#E13:150_3"><i><font color="Green">E306</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, <a class="ref" href="goboard7.html#T66">GOBOARD7:66</a></i>;<br/><a NAME="E9:150_3_7_2_1_2"/><b>then </b><i><font color="Green">E621</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:150_3_7_2_1_2"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="jordan.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E11:150_3_7_2_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan.html#E16:150_3"><i><font color="Green">E307</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E12:150_3_7_2_1_2"/><b>then </b><i><font color="Green">E622</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E13:150_3_7_2_1_2"/><i><font color="Green">E623</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pdx</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Palpha</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pnc</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E14:150_3_7_2_1_2"/><b>then </b><i><font color="Green">E624</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="txt" href="jordan.html#E52:150"><i><font color="Green">E261</font></i></a>, <a class="txt" href="jordan.html#E59:150"><i><font color="Green">E273</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E18:150_3"><i><font color="Green">E309</font></i></a>, <a class="ref" href="xboole_1.html#T114">XBOOLE_1:114</a></i>;<br/><a NAME="E15:150_3_7_2_1_2"/><i><font color="Green">E625</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E16:150_3_7_2_1_2"/><i><font color="Green">E626</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E17:150_3"><i><font color="Green">E308</font></i></a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:150_3_7_2_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E20:150_3"><i><font color="Green">E310</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, , , , <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E23:150_3"><i><font color="Green">E313</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">v</font> =  <a href="borsuk_2.html#K2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i><a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E20:150_3_7_2_1_2"/><i><font color="Green">E627</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/><a NAME="E21:150_3_7_2_1_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E22:150_3_7_2_1_2"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJc</font> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E23:150_3_7_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E24:150_3"><i><font color="Green">E314</font></i></a>, <a class="txt" href="jordan.html#E66:150"><i><font color="Green">E276</font></i></a>, <a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, <a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_2_1_3"><b>suppose </b></a><a NAME="E1:150_3_7_2_1_3"/><i><font color="Green">E628</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pnc</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fnc</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:150_3_7_2_1_3"/><i><font color="Green">E629</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fnc</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pnc</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fnc</font> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font>;<br/><a NAME="E4:150_3_7_2_1_3"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E5:150_3_7_2_1_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="ref" href="sppol_1.html#D2">SPPOL_1:def 2</a></i>;<br/><a NAME="E6:150_3_7_2_1_3"/><b>then </b><i><font color="Green">E630</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V1">horizontal</a>
 <b>by </b><i>, , <a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a></i>;<br/><a NAME="E7:150_3_7_2_1_3"/><i><font color="Green">E631</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E8:150_3_7_2_1_3"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, , , <a class="ref" href="jgraph_6.html#T11">JGRAPH_6:11</a></i>;<br/><a NAME="E9:150_3_7_2_1_3"/><b>then </b><i><font color="Green">E632</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E13:150_3"><i><font color="Green">E306</font></i></a>, , <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, <a class="ref" href="goboard7.html#T66">GOBOARD7:66</a></i>;<br/><a NAME="E10:150_3_7_2_1_3"/><b>then </b><i><font color="Green">E633</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:150_3_7_2_1_3"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="jordan.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E12:150_3_7_2_1_3"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan.html#E16:150_3"><i><font color="Green">E307</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E13:150_3_7_2_1_3"/><b>then </b><i><font color="Green">E634</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E14:150_3_7_2_1_3"/><i><font color="Green">E635</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pdx</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Palpha</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pnc</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:150_3_7_2_1_3"/><b>then </b><i><font color="Green">E636</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="txt" href="jordan.html#E52:150"><i><font color="Green">E261</font></i></a>, <a class="txt" href="jordan.html#E59:150"><i><font color="Green">E273</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E18:150_3"><i><font color="Green">E309</font></i></a>, <a class="ref" href="xboole_1.html#T114">XBOOLE_1:114</a></i>;<br/><a NAME="E16:150_3_7_2_1_3"/><i><font color="Green">E637</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E17:150_3_7_2_1_3"/><i><font color="Green">E638</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E17:150_3"><i><font color="Green">E308</font></i></a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:150_3_7_2_1_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E20:150_3"><i><font color="Green">E310</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, , , , <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E23:150_3"><i><font color="Green">E313</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">v</font> =  <a href="borsuk_2.html#K2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i><a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E21:150_3_7_2_1_3"/><i><font color="Green">E639</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/><a NAME="E22:150_3_7_2_1_3"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E23:150_3_7_2_1_3"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJc</font> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnc</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E24:150_3_7_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E24:150_3"><i><font color="Green">E314</font></i></a>, <a class="txt" href="jordan.html#E66:150"><i><font color="Green">E276</font></i></a>, <a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, <a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_3_7_2_1_4"><b>suppose </b></a><a NAME="E1:150_3_7_2_1_4"/><i><font color="Green">E640</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">Pnpg</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fnpg</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p4">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:150_3_7_2_1_4"/><i><font color="Green">E643</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fnpg</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pnpg</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fnpg</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Ppgc</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fpgc</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="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:150_3_7_2_1_4"/><i><font color="Green">E646</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fpgc</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Ppgc</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fpgc</font> )
 <b>by </b><i/>;<br/><a NAME="E6:150_3_7_2_1_4"/><i><font color="Green">E647</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="txt" href="jordan.html#E17:150"><i><font color="Green">E121</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><a NAME="E7:150_3_7_2_1_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V2">vertical</a>
 <b>by </b><i>, , <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/><a NAME="E8:150_3_7_2_1_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E13:150_3"><i><font color="Green">E306</font></i></a>, , , , <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/><a NAME="E9:150_3_7_2_1_4"/><b>then </b><i><font color="Green">E648</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppgc</font>;<br/><a NAME="E10:150_3_7_2_1_4"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="txt" href="jordan.html#E6:150_3"><i><font color="Green">E303</font></i></a>, <a class="txt" href="jordan.html#E50"><i><font color="Green">Lemma146</font></i></a></i>;<br/><a NAME="E11:150_3_7_2_1_4"/><b>then </b><i><font color="Green">E649</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E12:150_3_7_2_1_4"/><i><font color="Green">E650</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppgc</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pdx</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Palpha</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pnpg</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Ppgc</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:150_3_7_2_1_4"/><b>then </b><i><font color="Green">E651</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppgc</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Jc</font>
 <b>by </b><i><a class="txt" href="jordan.html#E52:150"><i><font color="Green">E261</font></i></a>, <a class="txt" href="jordan.html#E59:150"><i><font color="Green">E273</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E17:150_3"><i><font color="Green">E308</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, , </i>;<br/><a NAME="E14:150_3_7_2_1_4"/><i><font color="Green">E652</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppgc</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppgc</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:150_3_7_2_1_4"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E16:150_3"><i><font color="Green">E307</font></i></a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:150_3_7_2_1_4"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppgc</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan.html#E18:150_3"><i><font color="Green">E309</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, , , , <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="txt" href="jordan.html#E7:150_3"><i><font color="Green">E304</font></i></a>, <a class="txt" href="jordan.html#E10:150_3"><i><font color="Green">E305</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">v</font> =  <a href="borsuk_2.html#K2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppgc</font></span>)</span> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i><a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E19:150_3_7_2_1_4"/><i><font color="Green">E653</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/><a NAME="E20:150_3_7_2_1_4"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E21:150_3_7_2_1_4"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">PJc</font> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pdx</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Palpha</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pnpg</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Ppgc</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E22:150_3_7_2_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E22:150_3"><i><font color="Green">E312</font></i></a>, <a class="txt" href="jordan.html#E66:150"><i><font color="Green">E276</font></i></a>, <a class="txt" href="jordan.html#E20:150_3"><i><font color="Green">E310</font></i></a>, <a class="txt" href="jordan.html#E21:150_3"><i><font color="Green">E311</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon">V</font> be   <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="E90:150"/><i><font color="Green">E654</font></i>: 
<font color="Maroon">V</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font>
 ;<br/>

<b>assume </b><a NAME="E91:150"/><i><font color="Green">E655</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Ux</font>
 ;<br/>

<b>consider </b><font color="Maroon">VP</font> being   <a href="struct_0.html#NM2">Subset</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"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E93:150"/><i><font color="Green">E656</font></i>: 
<font color="Maroon">VP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E94:150"/><i><font color="Green">E657</font></i>: 
<font color="Maroon">VP</font> <a href="connsp_1.html#R3">is_a_component_of</a> <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"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E95:150"/>
<font color="Maroon">VP</font> is  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T17">JORDAN2C:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T2C</font> = <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"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2 ;<br/>
<a NAME="E97:150"/>
<font color="Maroon">VP</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</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"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T34">CONNSP_1:34</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">VP</font> = <font color="Maroon">VP</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2C</font> ;<br/>
<a NAME="E99:150"/><i><font color="Green">E658</font></i>: 
<font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<b>consider </b><font color="Maroon">Pjd</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fjd</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E101:150"/><i><font color="Green">E661</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fjd</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">Pjd</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fjd</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Plk</font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font>, <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span>, <font color="Maroon">flk</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="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E103:150"/><i><font color="Green">E664</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">flk</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> &amp; <font color="Maroon">Plk</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">flk</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">beta</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font>;<br/>
<a NAME="E104:150"/><i><font color="Green">E666</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span> <a href="hidden.html#R1">=</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"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pcm</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Plk</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pjd</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E105:150"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <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="E106:150"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span></span>)</span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="weierstr.html#T14">WEIERSTR:14</a></i>;<br/>
<a NAME="E107:150"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<a NAME="E108:150"/><b>then </b><i><font color="Green">E667</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a></i>;<br/>
<a NAME="E109:150"/><i><font color="Green">E668</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E42:150"><i><font color="Green">E199</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E110:150"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_4"><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="E1:150_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E2:150_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:150_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E111:150"/><b>then </b><i><font color="Green">E669</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E112:150"/><i><font color="Green">E670</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:150_5"/>
not <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:150_5"/><i><font color="Green">E671</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E4:150_5"/><i><font color="Green">E672</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:150_5"/>
<font color="Maroon">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Ux</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, , <a class="txt" href="jordan.html#E43:150"><i><font color="Green">E252</font></i></a>, <a class="txt" href="jordan.html#E44:150"><i><font color="Green">E253</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E6:150_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E24:150"><i><font color="Green">E182</font></i></a>, , , <a class="ref" href="jordan.html#T62" target="_self">Th62</a>, , <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E113:150"/>
( not  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not  <a href="jordan21.html#K1">UMP</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 
<b>by </b><i><a class="txt" href="jordan.html#E32:150"><i><font color="Green">E190</font></i></a>, , <a class="txt" href="jordan.html#E31:150"><i><font color="Green">E189</font></i></a>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E114:150"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T57">ZFMISC_1:57</a></i>;<br/>
<a NAME="E115:150"/><b>then </b><i><font color="Green">E673</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>
<a NAME="E116:150"/><i><font color="Green">E674</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="jordan.html#T61" target="_self">Th61</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E117:150"/><i><font color="Green">E675</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E118:150"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E119:150"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E43:150"><i><font color="Green">E252</font></i></a>, <a class="ref" href="xboole_1.html#T114">XBOOLE_1:114</a></i>;<br/>
<a NAME="E120:150"/><b>then </b><i><font color="Green">E676</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , , <a class="txt" href="jordan.html#E44:150"><i><font color="Green">E253</font></i></a>, <a class="txt" href="jordan.html#E45:150"><i><font color="Green">E254</font></i></a>, , <a class="ref" href="xboole_1.html#T114">XBOOLE_1:114</a></i>;<br/>
<a NAME="E121:150"/><i><font color="Green">E677</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</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"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <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/>
<a NAME="E122:150"/><i><font color="Green">E678</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></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"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <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/>
<a NAME="E123:150"/><i><font color="Green">E679</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</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"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span> <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/>
<a NAME="E124:150"/><i><font color="Green">E680</font></i>: 
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="txt" href="jordan.html#E39:150"><i><font color="Green">E195</font></i></a>, <a class="txt" href="jordan.html#E38:150"><i><font color="Green">E194</font></i></a>, , <a class="txt" href="jordan.html#E46:150"><i><font color="Green">E257</font></i></a>, <a class="txt" href="jordan.html#E34:150"><i><font color="Green">E192</font></i></a>, <a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E125:150"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E41:150"><i><font color="Green">E198</font></i></a>, <a class="ref" href="zfmisc_1.html#T55">ZFMISC_1:55</a></i>;<br/>
<a NAME="E126:150"/><b>then </b><i><font color="Green">E681</font></i>: 
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E50:150"><i><font color="Green">E259</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E127:150"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, , <a class="ref" href="jordan.html#D7" target="_self">Def7</a>, <a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E128:150"/><b>then </b><i><font color="Green">E682</font></i>: 
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E51:150"><i><font color="Green">E260</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E129:150"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T55">ZFMISC_1:55</a></i>;<br/>
<a NAME="E130:150"/><b>then </b><i><font color="Green">E683</font></i>: 
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E52:150"><i><font color="Green">E261</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E131:150"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan.html#D2" target="_self">Def2</a>, , <a class="ref" href="jordan.html#T64" target="_self">Th64</a>, <a class="txt" href="jordan.html#E47:150"><i><font color="Green">E258</font></i></a>, <a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E132:150"/><b>then </b>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E53:150"><i><font color="Green">E262</font></i></a>, , , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">ra</font> being  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E134:150"/><i><font color="Green">E685</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">ra</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E135:150"/><i><font color="Green">E686</font></i>: 
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E27:150"><i><font color="Green">E184</font></i></a>, <a class="txt" href="jordan.html#E28:150"><i><font color="Green">E185</font></i></a>, <a class="txt" href="jordan.html#E39:150"><i><font color="Green">E195</font></i></a>, <a class="txt" href="jordan.html#E38:150"><i><font color="Green">E194</font></i></a>, , <a class="txt" href="jordan.html#E46:150"><i><font color="Green">E257</font></i></a>, <a class="txt" href="jordan.html#E34:150"><i><font color="Green">E192</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E136:150"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E41:150"><i><font color="Green">E198</font></i></a>, <a class="ref" href="zfmisc_1.html#T55">ZFMISC_1:55</a></i>;<br/>
<a NAME="E137:150"/><b>then </b><i><font color="Green">E687</font></i>: 
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E55:150"><i><font color="Green">E264</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E138:150"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan.html#T60" target="_self">Th60</a>, <a class="txt" href="jordan.html#E37:150"><i><font color="Green">E193</font></i></a>, , <a class="ref" href="jordan.html#D7" target="_self">Def7</a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E139:150"/><b>then </b><i><font color="Green">E688</font></i>: 
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E56:150"><i><font color="Green">E265</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E140:150"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T55">ZFMISC_1:55</a></i>;<br/>
<a NAME="E141:150"/><b>then </b><i><font color="Green">E689</font></i>: 
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <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"><a href="topreal1.html#K3">LSeg</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E142:150"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan.html#D2" target="_self">Def2</a>, , <a class="ref" href="jordan.html#T64" target="_self">Th64</a>, <a class="txt" href="jordan.html#E47:150"><i><font color="Green">E258</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E143:150"/><b>then </b>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span>
 
<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">rb</font> being  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E145:150"/><i><font color="Green">E691</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">rb</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">ra</font>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">rb</font>;<br/>
<a NAME="E146:150"/><i><font color="Green">E692</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">ra</font>
 
<b>by </b><i/>;<br/>
<a NAME="E147:150"/><i><font color="Green">E693</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">rb</font>
 
<b>by </b><i/>;<br/>
<a NAME="E148:150"/>
not <font color="Maroon">VP</font> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E150:150"/><i><font color="Green">E694</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E151:150"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <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> : <font color="Olive">W</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E152:150"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E57:150"><i><font color="Green">E271</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E153:150"/><b>then </b><i><font color="Green">E695</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="txt" href="jordan.html#E7:150"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E154:150"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E29:150"><i><font color="Green">E186</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E155:150"/><b>then </b>
 <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">ra</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E157:150"/><i><font color="Green">E696</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">ra</font>
 <b>and </b><br/><a NAME="E158:150"/><i><font color="Green">E697</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E159:150"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E29:150"><i><font color="Green">E186</font></i></a>, <a class="txt" href="jordan.html#E58:150"><i><font color="Green">E272</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E160:150"/><b>then </b>
 <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">rb</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E162:150"/><i><font color="Green">E698</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">rb</font>
 <b>and </b><br/><a NAME="E163:150"/><i><font color="Green">E699</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font>, <font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E59:150"><i><font color="Green">E273</font></i></a>, <a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a></i>;<br/>
<a NAME="E165:150"/><i><font color="Green">E700</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">T2C</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">VP</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">VP</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font>, <font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">T2C</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">VP</font></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E61:150"><i><font color="Green">E274</font></i></a>, <a class="txt" href="jordan.html#E66:150"><i><font color="Green">E276</font></i></a>, , <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E167:150"/>
<font color="Maroon">T2C</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">VP</font> is <a href="borsuk_2.html#V1">arcwise_connected</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E168:150"/><b>then </b><i><font color="Green">E703</font></i>: 
<font color="Maroon">u1</font>,<font color="Maroon">v1</font> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D3">BORSUK_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">fuv</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"><font color="Maroon">T2C</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">VP</font></span>)</span><b> such that </b><br/><a NAME="E170:150"/><i><font color="Green">E705</font></i>: 
( <font color="Maroon">fuv</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">fuv</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">fuv</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/>
<a NAME="E171:150"/><i><font color="Green">E706</font></i>: 
<font color="Maroon">T2C</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">VP</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="goboard9.html#T4">GOBOARD9:4</a></i>;<br/>
<a NAME="E172:150"/>
<font color="Maroon">fuv</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">u1</font>,<font color="Maroon">v1</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">uv</font> = <font color="Maroon">fuv</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">u</font>,<font color="Maroon">v</font> <b>by </b><i><a class="txt" href="jordan.html#E72:150"><i><font color="Green">E278</font></i></a>, , <a class="ref" href="topalg_2.html#T1">TOPALG_2:1</a></i>;<br/>
<a NAME="E174:150"/><i><font color="Green">E708</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">fuv</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">T2C</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">VP</font></span>)</span>
 
;<br/>
<a NAME="E175:150"/><i><font color="Green">E709</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">uv</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="jordan.html#E71:150"><i><font color="Green">E277</font></i></a>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>consider </b><font color="Maroon">au</font> being    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">u</font>, <font color="Maroon">fau</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="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">u</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E177:150"/><i><font color="Green">E712</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fau</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">u</font> &amp; <font color="Maroon">au</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fau</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">vb</font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">v</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>, <font color="Maroon">fvb</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="topreal1.html#K3">LSeg</a> <font color="Maroon">v</font>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E179:150"/><i><font color="Green">E715</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">fvb</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">v</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">vb</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">fvb</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">AB</font> = <span class="p1">(<span class="default"><font color="Maroon">au</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">uv</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">vb</font>;<br/>
<a NAME="E180:150"/><i><font color="Green">E717</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">au</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">uv</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">vb</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">au</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">uv</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">vb</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E181:150"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">ra</font>
 
<b>by </b><i/>;<br/>
<a NAME="E182:150"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">u</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">ra</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E59:150"><i><font color="Green">E273</font></i></a>, <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E183:150"/><b>then </b><i><font color="Green">E718</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E54:150"><i><font color="Green">E263</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E184:150"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">rb</font>
 
<b>by </b><i/>;<br/>
<a NAME="E185:150"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">v</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">rb</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E64:150"><i><font color="Green">E275</font></i></a>, <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E186:150"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">v</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E187:150"/><b>then </b><i><font color="Green">E719</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">au</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">uv</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">vb</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="xboole_1.html#T114">XBOOLE_1:114</a></i>;<br/>
<a NAME="E188:150"/><i><font color="Green">E720</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D3">BORSUK_2:def 3</a></i>;<br/>
<a NAME="E189:150"/><i><font color="Green">E721</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, , <a class="ref" href="jordan.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E190:150"/><i><font color="Green">E722</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">u</font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E61:150"><i><font color="Green">E274</font></i></a>, <a class="txt" href="jordan.html#E29:150"><i><font color="Green">E186</font></i></a>, <a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, , <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E191:150"/><i><font color="Green">E723</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">v</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E66:150"><i><font color="Green">E276</font></i></a>, , <a class="txt" href="jordan.html#E29:150"><i><font color="Green">E186</font></i></a>, <a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E192:150"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">uv</font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, , <a class="txt" href="jordan.html#E71:150"><i><font color="Green">E277</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E193:150"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">u</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">uv</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E194:150"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">au</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">uv</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">vb</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <span class="p1">(<span class="default"><font color="Maroon">au</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">uv</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">vb</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">AR</font>,<font color="Maroon">BR</font> <b>by </b><i>, </i>;<br/>
<a NAME="E196:150"/><i><font color="Green">E724</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D3">BORSUK_2:def 3</a></i>;<br/>
<a NAME="E197:150"/><i><font color="Green">E725</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E198:150"/><i><font color="Green">E726</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p5">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p5"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E77:150"><i><font color="Green">E293</font></i></a>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E199:150"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="jordan.html#T61" target="_self">Th61</a>, <a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E200:150"/><b>then </b><i><font color="Green">E727</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pml</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p0">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p0"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Pkj</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E78:150"><i><font color="Green">E294</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E201:150"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font></span>)</span>,<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> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E26:150"><i><font color="Green">E183</font></i></a>, , <a class="ref" href="jordan.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>
<a NAME="E202:150"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span> <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="topreala.html#K1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E81:150"><i><font color="Green">E295</font></i></a>, , , , , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">CR</font>,<font color="Maroon">DR</font> <b>by </b><i><a class="txt" href="jordan.html#E75:150"><i><font color="Green">E292</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E205:150"/><i><font color="Green">E728</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:150"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:150"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jgraph_8.html#T6">JGRAPH_8:6</a></i>;<br/>
<a NAME="E206:150"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</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="E207:150"/><b>then </b>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">au</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">uv</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">vb</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Pcm</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pml</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Plk</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pkj</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Pjd</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E208:150"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E82:150"><i><font color="Green">E296</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div>
