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<div class="add">

<b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">d</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c11">p1</font>, <font color="Maroon" title="c12">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c13">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>













<b>set </b><font color="Maroon" title="c14">R</font> =  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>;<br/>
<b>set </b><font color="Maroon" title="c15">dR</font> =  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>;<br/>
<b>set </b><font color="Maroon" title="c16">n</font> =  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:85"/><i><font color="Green" title="E58">A1</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c9">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">d</font> )
 <b>and </b><br/><a NAME="E2:85"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 <b>and </b><br/><a NAME="E3:85"/><i><font color="Green" title="E60">A3</font></i>: 
not <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 <b>and </b><br/><a NAME="E4:85"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c13">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E5:85"/><i><font color="Green" title="E62">A5</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:85"/><i><font color="Green" title="E63">A6</font></i>: 
not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E5:85"><i><font color="Green" title="E62">A5</font></i></a></i>;<br/>
<a NAME="E8:85"/><i><font color="Green" title="E64">A7</font></i>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:85"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="jordan.html#T52" target="_self" title="JORDAN:th.52">Th52</a></i>;<br/>
<a NAME="E9:85"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c13">P</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="jordan6.html#T11" title="JORDAN6:th.11">JORDAN6:11</a></i>;<br/>
<a NAME="E10:85"/>
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E11:85"/><b>then </b><i><font color="Green" title="E66">A9</font></i>: 
<font color="Maroon" title="c13">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E2:85"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E12:85"/>
<font color="Maroon" title="c12">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E13:85"/><b>then </b>
<font color="Maroon" title="c13">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E3:85"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:85"/><b>then </b><i><font color="Green" title="E67">A10</font></i>: 
<font color="Maroon" title="c13">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:85"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="jordan.html#E9:85"><i><font color="Green" title="E65">A8</font></i></a>, <a class="txt" href="jordan.html#E11:85"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="connsp_1.html#T23" title="CONNSP_1:th.23">CONNSP_1:23</a></i>;<br/>
<a NAME="E15:85"/>
<font color="Maroon" title="c13">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="jordan6.html#T12" title="JORDAN6:th.12">JORDAN6:12</a></i>;<br/>
<a NAME="E16:85"/><b>then </b><i><font color="Green" title="E68">A11</font></i>: 
<font color="Maroon" title="c13">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35" title="TOPS_1:th.35">TOPS_1:35</a></i>;<br/>
<a NAME="E17:85"/><b>then </b><i><font color="Green" title="E69">A12</font></i>: 
 <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E14:85"><i><font color="Green" title="E67">A10</font></i></a>, <a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="jordan5c.html#D1" title="JORDAN5C:def.1">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" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">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" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:85_1_1"/><b>then </b><i><font color="Green" title="E70">A13</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="txt" href="jordan.html#E17:85"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:85_1_1"/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:85"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="jordan.html#T45" target="_self" title="JORDAN:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E4:85_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E2:85_1_1"><i><font color="Green" title="E70">A13</font></i></a>, <a class="txt" href="jordan.html#E6:85"><i><font color="Green" title="E63">A6</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" title="E70">A14</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c19">P</font> = <font color="Maroon" title="c13">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">P</font></span>)</span><b> such that </b><br/><a NAME="E4:85_1_2"/><i><font color="Green" title="E71">A15</font></i>: 
<font color="Maroon" title="c20">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E5:85_1_2"/><i><font color="Green" title="E72">A16</font></i>: 
( <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p2</font> )
 <b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><i><font color="Green" title="E73">A17</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c20">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">P</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E4:85_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c19">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E7:85_1_2"/>
 <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">P</font>
 <b>by </b><i><a class="txt" href="jordan.html#E17:85"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">na</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:85_1_2"/><i><font color="Green" title="E74">A18</font></i>: 
<font color="Maroon" title="c21">na</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">f</font>
 <b>and </b><br/><a NAME="E10:85_1_2"/><i><font color="Green" title="E75">A19</font></i>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">na</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E6:85_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E11:85_1_2"/>
<font color="Maroon" title="c21">na</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 <b>by </b><i><a class="txt" href="jordan.html#E9:85_1_2"><i><font color="Green" title="E74">A18</font></i></a>, <a class="ref" href="membered.html#D2" title="MEMBERED:def.2">MEMBERED:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">na</font> = <font color="Maroon" title="c21">na</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E13:85_1_2"/><i><font color="Green" title="E76">A20</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">na</font> &amp; <font color="Maroon" title="c22">na</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan.html#E9:85_1_2"><i><font color="Green" title="E74">A18</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/><a NAME="E14:85_1_2"/><i><font color="Green" title="E77">A21</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">P</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">xa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:85_1_2"/><i><font color="Green" title="E78">A22</font></i>: 
<font color="Maroon" title="c23">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">f</font>
 <b>and </b><br/><a NAME="E17:85_1_2"/><i><font color="Green" title="E79">A23</font></i>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="jordan.html#E6:85_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="txt" href="jordan.html#E5:85"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E18:85_1_2"/>
<font color="Maroon" title="c23">xa</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 <b>by </b><i><a class="txt" href="jordan.html#E16:85_1_2"><i><font color="Green" title="E78">A22</font></i></a>, <a class="ref" href="membered.html#D2" title="MEMBERED:def.2">MEMBERED:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">xa</font> = <font color="Maroon" title="c23">xa</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E20:85_1_2"/><i><font color="Green" title="E80">A24</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">xa</font> &amp; <font color="Maroon" title="c24">xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan.html#E16:85_1_2"><i><font color="Green" title="E78">A22</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/><a NAME="E21:85_1_2"/><i><font color="Green" title="E81">A25</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="jordan.html#E14:85_1_2"><i><font color="Green" title="E77">A21</font></i></a>, <a class="txt" href="jordan.html#E2:85"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="jordan.html#E6:85"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="jordan.html#E5:85"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="jordan16.html#T39" title="JORDAN16:th.39">JORDAN16:39</a></i>;<br/><a NAME="E22:85_1_2"/><b>then </b><i><font color="Green" title="E82">A26</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T11" title="JORDAN6:th.11">JORDAN6:11</a></i>;<br/><a NAME="E23:85_1_2"/>
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="jordan.html#E21:85_1_2"><i><font color="Green" title="E81">A25</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/><a NAME="E24:85_1_2"/><b>then </b><i><font color="Green" title="E83">A27</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="txt" href="jordan.html#E2:85"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E25:85_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="jordan.html#E21:85_1_2"><i><font color="Green" title="E81">A25</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">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" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E6:85"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E27:85_1_2"/><b>then </b>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E22:85_1_2"><i><font color="Green" title="E82">A26</font></i></a>, <a class="txt" href="jordan.html#E24:85_1_2"><i><font color="Green" title="E83">A27</font></i></a>, <a class="ref" href="connsp_1.html#T23" title="CONNSP_1:th.23">CONNSP_1:23</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E29:85_1_2"/><i><font color="Green" title="E84">A28</font></i>: 
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>
 <b>and </b><br/><a NAME="E30:85_1_2"/><i><font color="Green" title="E85">A29</font></i>: 
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:85"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">z</font> = <font color="Maroon" title="c25">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E29:85_1_2"><i><font color="Green" title="E84">A28</font></i></a></i>;<br/><a NAME="E32:85_1_2"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c11">p1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">zz</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E34:85_1_2"/><i><font color="Green" title="E86">A30</font></i>: 
<font color="Maroon" title="c27">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">z</font>
 <b>and </b><br/><a NAME="E35:85_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c27">zz</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>
 <b>and </b><br/><a NAME="E36:85_1_2"/><i><font color="Green" title="E87">A31</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c27">zz</font>,<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>
 <b>by </b><i><a class="txt" href="jordan.html#E29:85_1_2"><i><font color="Green" title="E84">A28</font></i></a></i>;<br/><a NAME="E37:85_1_2"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">P</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c28">za</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E39:85_1_2"/><i><font color="Green" title="E88">A32</font></i>: 
<font color="Maroon" title="c28">za</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">f</font>
 <b>and </b><br/><a NAME="E40:85_1_2"/><i><font color="Green" title="E89">A33</font></i>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">za</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">z</font>
 <b>by </b><i><a class="txt" href="jordan.html#E6:85_1_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="txt" href="jordan.html#E29:85_1_2"><i><font color="Green" title="E84">A28</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E41:85_1_2"/>
<font color="Maroon" title="c28">za</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 <b>by </b><i><a class="txt" href="jordan.html#E39:85_1_2"><i><font color="Green" title="E88">A32</font></i></a>, <a class="ref" href="membered.html#D2" title="MEMBERED:def.2">MEMBERED:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">za</font> = <font color="Maroon" title="c28">za</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E43:85_1_2"/><i><font color="Green" title="E90">A34</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">za</font> &amp; <font color="Maroon" title="c29">za</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan.html#E39:85_1_2"><i><font color="Green" title="E88">A32</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/><a NAME="E44:85_1_2"/><i><font color="Green" title="E91">A35</font></i>: 
<font color="Maroon" title="c22">na</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">za</font>
 <b>by </b><i><a class="txt" href="jordan.html#E30:85_1_2"><i><font color="Green" title="E85">A29</font></i></a>, <a class="txt" href="jordan.html#E40:85_1_2"><i><font color="Green" title="E89">A33</font></i></a>, <a class="txt" href="jordan.html#E43:85_1_2"><i><font color="Green" title="E90">A34</font></i></a>, <a class="txt" href="jordan.html#E10:85_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="jordan.html#E13:85_1_2"><i><font color="Green" title="E76">A20</font></i></a>, <a class="txt" href="jordan.html#E14:85"><i><font color="Green" title="E67">A10</font></i></a>, <a class="txt" href="jordan.html#E16:85"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="jordan.html#E4:85_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="jordan.html#E5:85_1_2"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="jordan.html#E4:85"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="jordan5c.html#D1" title="JORDAN5C:def.1">JORDAN5C:def 1</a></i>;<br/><a NAME="E45:85_1_2"/><i><font color="Green" title="E92">A36</font></i>: 
<font color="Maroon" title="c29">za</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">xa</font>
 <b>by </b><i><a class="txt" href="jordan.html#E4:85_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="jordan.html#E5:85_1_2"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="jordan.html#E36:85_1_2"><i><font color="Green" title="E87">A31</font></i></a>, <a class="txt" href="jordan.html#E34:85_1_2"><i><font color="Green" title="E86">A30</font></i></a>, <a class="txt" href="jordan.html#E40:85_1_2"><i><font color="Green" title="E89">A33</font></i></a>, <a class="txt" href="jordan.html#E43:85_1_2"><i><font color="Green" title="E90">A34</font></i></a>, <a class="txt" href="jordan.html#E17:85_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="txt" href="jordan.html#E20:85_1_2"><i><font color="Green" title="E80">A24</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><a NAME="E46:85_1_2"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c11">p1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>, <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">xx</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E48:85_1_2"/><i><font color="Green" title="E93">A37</font></i>: 
<font color="Maroon" title="c30">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font>
 <b>and </b><br/><a NAME="E49:85_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c30">xx</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>
 <b>and </b><br/><a NAME="E50:85_1_2"/><i><font color="Green" title="E94">A38</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c30">xx</font>, <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span>)</span>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>
 <b>by </b><i><a class="txt" href="jordan.html#E5:85"><i><font color="Green" title="E62">A5</font></i></a></i>;<br/><a NAME="E51:85_1_2"/>
<font color="Maroon" title="c24">xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">na</font>
 <b>by </b><i><a class="txt" href="jordan.html#E4:85_1_2"><i><font color="Green" title="E71">A15</font></i></a>, <a class="txt" href="jordan.html#E5:85_1_2"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="jordan.html#E50:85_1_2"><i><font color="Green" title="E94">A38</font></i></a>, <a class="txt" href="jordan.html#E48:85_1_2"><i><font color="Green" title="E93">A37</font></i></a>, <a class="txt" href="jordan.html#E17:85_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="txt" href="jordan.html#E20:85_1_2"><i><font color="Green" title="E80">A24</font></i></a>, <a class="txt" href="jordan.html#E10:85_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="jordan.html#E13:85_1_2"><i><font color="Green" title="E76">A20</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><a NAME="E52:85_1_2"/><b>then </b>
<font color="Maroon" title="c24">xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c22">na</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:85_1_2"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="jordan.html#E10:85_1_2"><i><font color="Green" title="E75">A19</font></i></a>, <a class="txt" href="jordan.html#E17:85_1_2"><i><font color="Green" title="E79">A23</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E53:85_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E44:85_1_2"><i><font color="Green" title="E91">A35</font></i></a>, <a class="txt" href="jordan.html#E45:85_1_2"><i><font color="Green" title="E92">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
