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<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:124"/><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>
 <b>and </b><br/><a NAME="E2:124"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <b>and </b><br/><a NAME="E3:124"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</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/>

<a NAME="E4:124"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">C</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> <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>by </b><i><a class="txt" href="jordan.html#E52"><i><font color="Green">Lemma149</font></i></a>, <a class="ref" href="jordan.html#T33" target="_self">Th33</a></i>;<br/>
<b>assume </b><a NAME="E5:124"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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#R2">meets</a> <font color="Maroon">C</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="E7:124"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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>
 <b>and </b><br/><a NAME="E8:124"/><i><font color="Green">E80</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="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/>;<br/>
<a NAME="E10:124"/>
<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">1,0</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>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E11:124"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">p</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">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#E53"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="jordan.html#T22" target="_self">Th22</a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E12:124"/><i><font color="Green">E82</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#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E53"><i><font color="Green">Lemma151</font></i></a>, , <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E13:124"/><i><font color="Green">E83</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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="E14:124"/>
<font color="Maroon">p</font> <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">1,0</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#E53"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="jordan.html#T10" target="_self">Th10</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E15:124"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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="ref" href="jordan.html#T43" target="_self">Th43</a>, <a class="ref" href="jordan.html#T44" target="_self">Th44</a>, <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, , <a class="ref" href="jordan.html#T10" target="_self">Th10</a>, , <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/>
<a NAME="E16:124"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<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/>
<a NAME="E17:124"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="jordan.html#T42" target="_self">Th42</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E18:124"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="jordan.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E19:124"/>
<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">1,0</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>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E20:124"/><b>then </b><i><font color="Green">E110</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="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E53"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="jordan.html#T22" target="_self">Th22</a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E21:124"/><i><font color="Green">E115</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> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E52"><i><font color="Green">Lemma149</font></i></a>, <a class="ref" href="jordan24.html#D1">JORDAN24:def 1</a></i>;<br/>
<a NAME="E22:124"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E23:124"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<a NAME="E24:124"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">p</font> <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">1,0</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#E53"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="jordan.html#T10" target="_self">Th10</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E25:124"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<b>hence </b><a NAME="E26:124"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan.html#E12:124"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan.html#T42" target="_self">Th42</a>, , , <a class="ref" href="jordan.html#T43" target="_self">Th43</a>, <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>


</div>
