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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be   <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>set </b><font color="Maroon">R</font> =  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<b>set </b><font color="Maroon">dR</font> =  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<b>set </b><font color="Maroon">n</font> =  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:85"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> )
 <b>and </b><br/><a NAME="E2:85"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E3:85"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E4:85"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<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><b>that </b><br/><a NAME="E5:85"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:85"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</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#E2"><i><font color="Green">Lemma62</font></i></a></i>;<br/>
<a NAME="E8:85"/><i><font color="Green">E81</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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:85"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">P</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="E10:85"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E11:85"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E12:85"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E13:85"/><b>then </b>
<font color="Maroon">P</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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></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#E1"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:85"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 
<b>by </b><i>, , , <a class="ref" href="connsp_1.html#T23">CONNSP_1:23</a></i>;<br/>
<a NAME="E15:85"/>
<font color="Maroon">P</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="E16:85"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></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="E17:85"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="jordan5c.html#D1">JORDAN5C:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1_1"><b>suppose </b></a><a NAME="E1:85_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:85_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:85_1_1"/>
 <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:85_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E2:85"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan.html#E3"><i><font color="Green">Lemma63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1_2"><b>suppose </b></a><a NAME="E1:85_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as  non <a href="xboole_0.html#V1">empty</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> <b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/><b>consider </b><font color="Maroon">f</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">P</font></span>)</span><b> such that </b><br/><a NAME="E4:85_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E5:85_1_2"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><i><font color="Green">E121</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T2" target="_self">Th2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E7:85_1_2"/>
 <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:85"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">na</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:85_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">na</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E10:85_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">na</font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E11:85_1_2"/>
<font color="Maroon">na</font> is <a href="xreal_0.html#V1">real</a>
 <b>by </b><i><a class="txt" href="jordan.html#E3:85"><i><font color="Green">E74</font></i></a>, <a class="ref" href="membered.html#D2">MEMBERED:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">na</font> = <font color="Maroon">na</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E13:85_1_2"/><i><font color="Green">E177</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">na</font> &amp; <font color="Maroon">na</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan.html#E3:85"><i><font color="Green">E74</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E14:85_1_2"/><i><font color="Green">E178</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5">JORDAN16:5</a></i>;<br/><b>then consider </b><font color="Maroon">xa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:85_1_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">xa</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E17:85_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="jordan.html#T4" target="_self">Th4</a>, <a class="txt" href="jordan.html#E2"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E18:85_1_2"/>
<font color="Maroon">xa</font> is <a href="xreal_0.html#V1">real</a>
 <b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green">E78</font></i></a>, <a class="ref" href="membered.html#D2">MEMBERED:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">xa</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E20:85_1_2"/><i><font color="Green">E182</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">xa</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green">E78</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E21:85_1_2"/><i><font color="Green">E183</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="txt" href="jordan.html#E3"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="jordan.html#E2"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="jordan16.html#T39">JORDAN16:39</a></i>;<br/><a NAME="E22:85_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</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="E23:85_1_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:85"><i><font color="Green">E81</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/><a NAME="E24:85_1_2"/><b>then </b><i><font color="Green">E185</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E25:85_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:85"><i><font color="Green">E81</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/><a NAME="E26:85_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></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#E3"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E27:85_1_2"/><b>then </b>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <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> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E9:85"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E11:85"><i><font color="Green">E83</font></i></a>, <a class="ref" href="connsp_1.html#T23">CONNSP_1:23</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E29:85_1_2"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E30:85_1_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</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">z</font> = <font color="Maroon">z</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#E13"><i><font color="Green">Lemma84</font></i></a></i>;<br/><a NAME="E32:85_1_2"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Olive">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">x</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then consider </b><font color="Maroon">zz</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E34:85_1_2"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E35:85_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">zz</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E36:85_1_2"/><i><font color="Green">E190</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">zz</font>,<font color="Maroon">x</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a></i>;<br/><a NAME="E37:85_1_2"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5">JORDAN16:5</a></i>;<br/><b>then consider </b><font color="Maroon">za</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E39:85_1_2"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">za</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E40:85_1_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">za</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="jordan.html#T4" target="_self">Th4</a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E41:85_1_2"/>
<font color="Maroon">za</font> is <a href="xreal_0.html#V1">real</a>
 <b>by </b><i><a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="membered.html#D2">MEMBERED:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">za</font> = <font color="Maroon">za</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E43:85_1_2"/><i><font color="Green">E194</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">za</font> &amp; <font color="Maroon">za</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E44:85_1_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">na</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">za</font>
 <b>by </b><i><a class="ref" href="jordan.html#T10" target="_self">Th10</a>, <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan.html#T8" target="_self">Th8</a>, , , <a class="ref" href="jordan.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan.html#T3" target="_self">Th3</a>, , <a class="ref" href="jordan5c.html#D1">JORDAN5C:def 1</a></i>;<br/><a NAME="E45:85_1_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">za</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xa</font>
 <b>by </b><i><a class="ref" href="jordan.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan.html#T3" target="_self">Th3</a>, , , <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="txt" href="jordan.html#E5:85"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan.html#E6:85"><i><font color="Green">E80</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><a NAME="E46:85_1_2"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Olive">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p</font>, <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then consider </b><font color="Maroon">xx</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E48:85_1_2"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E49:85_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">xx</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E50:85_1_2"/><i><font color="Green">E199</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">xx</font>, <a href="jordan5c.html#K1">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan.html#E2"><i><font color="Green">Lemma62</font></i></a></i>;<br/><a NAME="E51:85_1_2"/>
<font color="Maroon">xa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">na</font>
 <b>by </b><i><a class="ref" href="jordan.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan.html#T3" target="_self">Th3</a>, , <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="txt" href="jordan.html#E5:85"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan.html#E6:85"><i><font color="Green">E80</font></i></a>, <a class="ref" href="jordan.html#T8" target="_self">Th8</a>, , <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><a NAME="E52:85_1_2"/><b>then </b>
<font color="Maroon">xa</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">na</font>
 <b>by </b><i><a class="txt" href="jordan.html#E2:85"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan.html#E5:85"><i><font color="Green">E79</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E53:85_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
