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

<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p4</font>, <font color="Maroon" title="c5">p</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>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green" title="E37">A1</font></i>: 
( ( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) or ( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green" title="E38">A2</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E3:38"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan12.html#E2:38"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:38"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E5:38"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>
 
<b>by </b><i><a class="txt" href="jordan12.html#E3:38"><i><font color="Green" title="E39">A3</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="E6:38"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">p4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T11" title="TOPREAL1:th.11">TOPREAL1:11</a></i>;<br/>
<a NAME="E7:38"/><b>then </b><i><font color="Green" title="E41">A5</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:38"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E9:38"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E10:38"/><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<div><i><font color="Green" title="E45">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 ;<br/><a NAME="E2:38_1"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> )
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E4:38"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="jordan12.html#T25" target="_self" title="JORDAN12:th.25">Th25</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E46">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2"/>
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><a NAME="E2:38_2"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> )
 <b>by </b><i><a class="ref" href="sppol_1.html#T36" title="SPPOL_1:th.36">SPPOL_1:36</a></i>;<br/><b>hence </b><a NAME="E3:38_2"/>
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan12.html#E4:38"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="sprect_3.html#T21" title="SPRECT_3:th.21">SPRECT_3:21</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_3"/><i><font color="Green" title="E47">A11</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">p1</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> )
 ;<br/><a NAME="E2:38_3"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E3:38"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><i><font color="Green" title="E49">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_3_1"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 ;<br/><a NAME="E2:38_3_1"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E10:38"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:38_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan12.html#E2:38"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="jordan12.html#E1:38_3"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_3_2_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">p</font> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">p</font> )
 </span><b>by </b><i><a class="txt" href="jordan12.html#E1:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jordan12.html#E11:38"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="jordan12.html#E12:38"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="jordan12.html#E1:38_3"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="jordan12.html#E2:38_3"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="jordan12.html#E3:38_3"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="jordan12.html#T29" target="_self" title="JORDAN12:th.29">Th29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_3_2_1_1"><b>suppose </b></a><a NAME="E1:38_3_2_1_1"/>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">p</font>
 ;<br/><div class="add"><a NAME="E2:38_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E7:38"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="jordan12.html#E8:38"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:38_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan12.html#E2:38"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="jordan12.html#E1:38_3"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_3_2_1_2"><b>suppose </b></a><a NAME="E1:38_3_2_1_2"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">p</font>
 ;<br/><div class="add"><a NAME="E2:38_3_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan12.html#E7:38"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="jordan12.html#E9:38"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:38_3_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan12.html#E2:38"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="jordan12.html#E1:38_3"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:38_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:38"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p1</font> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> )
 ;<br/>


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