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<b>let </b><font color="Maroon">c<sub>22</sub></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<sub>23</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:122"/><i><font color="Green">E147</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<sub>23</sub></font>
 <b>and </b><br/><a NAME="E2:122"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>and </b><br/><a NAME="E3:122"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">c<sub>22</sub></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">1,3</span><a href="euclid.html#K23">]|</a></span>
 ;<br/>

<a NAME="E4:122"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">c<sub>23</sub></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#E1:122"><i><font color="Green">E147</font></i></a>, <a class="ref" href="jordan.html#T74" target="_self">Th74</a></i>;<br/>
<b>assume </b><a NAME="E5:122"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>22</sub></font>,<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#R2">meets</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:122"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>22</sub></font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,3</span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E8:122"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></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">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></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#E8:122"><i><font color="Green">E152</font></i></a></i>;<br/>
<a NAME="E10:122"/>
<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">1,0</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="E11:122"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">c<sub>22</sub></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">1,3</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#E3:122"><i><font color="Green">E149</font></i></a>, <a class="txt" href="jordan.html#E128"><i><font color="Green">Lemma111</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E12:122"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>22</sub></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">1,3</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#E3:122"><i><font color="Green">E149</font></i></a>, <a class="txt" href="jordan.html#E108"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E13:122"/><i><font color="Green">E155</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>22</sub></font>,<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="txt" href="jordan.html#E11:122"><i><font color="Green">E153</font></i></a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E14:122"/>
<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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E3:122"><i><font color="Green">E149</font></i></a>, <a class="txt" href="jordan.html#E98"><i><font color="Green">Lemma85</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E15:122"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>22</sub></font>,<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#E12:122"><i><font color="Green">E154</font></i></a>, <a class="txt" href="jordan.html#E13:122"><i><font color="Green">E155</font></i></a>, <a class="txt" href="jordan.html#E125"><i><font color="Green">Lemma108</font></i></a>, <a class="txt" href="jordan.html#E11:122"><i><font color="Green">E153</font></i></a>, <a class="txt" href="jordan.html#E98"><i><font color="Green">Lemma85</font></i></a>, <a class="txt" href="jordan.html#E107"><i><font color="Green">Lemma94</font></i></a>, <a class="txt" href="jordan.html#E109"><i><font color="Green">Lemma96</font></i></a>, <a class="txt" href="jordan.html#E110"><i><font color="Green">Lemma97</font></i></a>, <a class="ref" href="goboard7.html#T65">GOBOARD7:65</a></i>;<br/>
<a NAME="E16:122"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>22</sub></font>,<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="txt" href="jordan.html#E121"><i><font color="Green">Lemma105</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:122"/><b>then </b>
<font color="Maroon">c<sub>25</sub></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<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:122"><i><font color="Green">E152</font></i></a>, <a class="txt" href="jordan.html#E7:122"><i><font color="Green">E151</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E18:122"/><b>then </b><i><font color="Green">E156</font></i>: 
( <font color="Maroon">c<sub>25</sub></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">c<sub>25</sub></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="txt" href="jordan.html#E4:122"><i><font color="Green">E150</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E19:122"/>
<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">1,3</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="E20:122"/><b>then </b><i><font color="Green">E157</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">c<sub>22</sub></font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E3:122"><i><font color="Green">E149</font></i></a>, <a class="txt" href="jordan.html#E128"><i><font color="Green">Lemma111</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E21:122"/><i><font color="Green">E158</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<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:122"><i><font color="Green">E147</font></i></a>, <a class="ref" href="jordan24.html#D1">JORDAN24:def 1</a></i>;<br/>
<a NAME="E22:122"/>
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E2:122"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E23:122"/><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">c<sub>22</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E21:122"><i><font color="Green">E158</font></i></a>, <a class="txt" href="jordan.html#E20:122"><i><font color="Green">E157</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<a NAME="E24:122"/><b>then </b><i><font color="Green">E159</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#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E3:122"><i><font color="Green">E149</font></i></a>, <a class="txt" href="jordan.html#E98"><i><font color="Green">Lemma85</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>
<a NAME="E25:122"/>
<font color="Maroon">c<sub>22</sub></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">c<sub>22</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></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="E26:122"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E24:122"><i><font color="Green">E159</font></i></a>, <a class="txt" href="jordan.html#E18:122"><i><font color="Green">E156</font></i></a>, <a class="txt" href="jordan.html#E95"><i><font color="Green">Lemma82</font></i></a>, <a class="txt" href="jordan.html#E7:122"><i><font color="Green">E151</font></i></a>, <a class="txt" href="jordan.html#E107"><i><font color="Green">Lemma94</font></i></a>, <a class="txt" href="jordan.html#E11:122"><i><font color="Green">E153</font></i></a>, <a class="txt" href="jordan.html#E12:122"><i><font color="Green">E154</font></i></a>, <a class="txt" href="jordan.html#E114"><i><font color="Green">Lemma101</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>


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