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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="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="E1:136"/><i><font color="Green">E161</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="E2:136"/><i><font color="Green">E162</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>
 ;<br/>

<a NAME="E3:136"/><i><font color="Green">E163</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#E1:136"><i><font color="Green">E161</font></i></a>, <a class="ref" href="jordan.html#T74" target="_self">Th74</a></i>;<br/>
<a NAME="E4:136"/>
<font color="Maroon">c<sub>23</sub></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#E1:136"><i><font color="Green">E161</font></i></a>, <a class="ref" href="jordan.html#T71" target="_self">Th71</a></i>;<br/>
<a NAME="E5:136"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">c<sub>22</sub></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><a class="txt" href="jordan.html#E2:136"><i><font color="Green">E162</font></i></a></i>;<br/>
<a NAME="E6:136"/><i><font color="Green">E165</font></i>: 
ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 
<b>by </b><i><a class="txt" href="jordan.html#E5:136"><i><font color="Green">E164</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:136_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <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="E2:136_1"/><b>then </b>
<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"><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">1,3</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E6:136"><i><font color="Green">E165</font></i></a>, <a class="txt" href="jordan.html#E100"><i><font color="Green">Lemma87</font></i></a>, <a class="txt" href="jordan.html#E103"><i><font color="Green">Lemma90</font></i></a>, <a class="txt" href="jordan.html#E104"><i><font color="Green">Lemma91</font></i></a>, <a class="txt" href="jordan.html#E107"><i><font color="Green">Lemma94</font></i></a>, <a class="txt" href="jordan.html#E108"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="goboard7.html#T9">GOBOARD7:9</a></i>;<br/><a NAME="E3:136_1"/><b>then </b>
<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#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#E2:136"><i><font color="Green">E162</font></i></a>, <a class="txt" href="jordan.html#E119"><i><font color="Green">Lemma104</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:136_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E1:136_1"><i><font color="Green">E166</font></i></a>, <a class="txt" href="jordan.html#E97"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="jordan.html#E98"><i><font color="Green">Lemma85</font></i></a>, <a class="txt" href="jordan.html#E100"><i><font color="Green">Lemma87</font></i></a>, <a class="txt" href="jordan.html#E3:136"><i><font color="Green">E163</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:136"/>
<font color="Maroon">c<sub>22</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> 3
 <b>by </b><i><a class="txt" href="jordan.html#E6:136"><i><font color="Green">E165</font></i></a>, <a class="txt" href="jordan.html#E100"><i><font color="Green">Lemma87</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


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