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

<b>let </b><font color="Maroon" title="c22">P</font>, <font color="Maroon" title="c23">A</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>assume </b><b>that </b><br/><a NAME="E1:145"/><i><font color="Green" title="E170">A1</font></i>: 
<font color="Maroon" title="c22">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>and </b><br/><a NAME="E2:145"/><i><font color="Green" title="E171">A2</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Maroon" title="c22">P</font>
 <b>and </b><br/><a NAME="E3:145"/><i><font color="Green" title="E172">A3</font></i>: 
<font color="Maroon" title="c23">A</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c22">P</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c24">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="E4:145"/><i><font color="Green" title="E173">A4</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">A</font>
 <b>and </b><br/><a NAME="E5:145"/><i><font color="Green" title="E174">A5</font></i>: 
not <font color="Maroon" title="c24">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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3
 ;<br/>

<a NAME="E6:145"/>
<font color="Maroon" title="c22">P</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E2:145"><i><font color="Green" title="E171">A2</font></i></a>, <a class="ref" href="jordan.html#T71" target="_self" title="JORDAN:th.71">Th71</a></i>;<br/>
<a NAME="E7:145"/><b>then </b><i><font color="Green" title="E175">A6</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">x</font> = <font color="Maroon" title="c24">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#E4:145"><i><font color="Green" title="E173">A4</font></i></a></i>;<br/>
<a NAME="E9:145"/><i><font color="Green" title="E176">A7</font></i>: 
(  not  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Maroon" title="c25">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or  not  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Maroon" title="c25">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 )
 
<b>by </b><i><a class="txt" href="jordan.html#E5:145"><i><font color="Green" title="E174">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="145_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:145_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c25">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="145_1_1"><b>suppose </b></a><a NAME="E1:145_1_1"/><i><font color="Green" title="E177">A8</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c26">E</font> =  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c25">x</font>;<br/><a NAME="E2:145_1_1"/>
 <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c25">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">e</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><a NAME="E1:145_1_1_1"/><i><font color="Green" title="E178">A9</font></i>: 
<font color="Maroon" title="c27">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c25">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c28">e</font> = <font color="Maroon" title="c27">e</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> ;<br/>
<a NAME="E3:145_1_1_1"/><i><font color="Green" title="E179">A10</font></i>: 
( <font color="Maroon" title="c28">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c25">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">e</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="c25">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jordan.html#E1:145_1_1_1"><i><font color="Green" title="E178">A9</font></i></a>, <a class="ref" href="topreal1.html#D13" title="TOPREAL1:def.13">TOPREAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_1_1_1_1"/>
<font color="Maroon" title="c28">e</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3
 ;<br/><a NAME="E2:145_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">p</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> st <br/>( <font color="Maroon" title="c28">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 )
 ;<br/><b>hence </b><a NAME="E3:145_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E9:145"><i><font color="Green" title="E176">A7</font></i></a>, <a class="txt" href="jordan.html#E1:145_1_1"><i><font color="Green" title="E177">A8</font></i></a>, <a class="txt" href="jordan.html#E3:145_1_1_1"><i><font color="Green" title="E179">A10</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>
<b>hence </b><a NAME="E5:145_1_1_1"/>
<font color="Maroon" title="c27">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:145_1_1"/><b>then </b>
 <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c25">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E7:145"><i><font color="Green" title="E175">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:145_1_1"/><b>then </b>
 <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c25">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/><a NAME="E5:145_1_1"/><b>then </b><i><font color="Green" title="E178">A11</font></i>: 
 <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c25">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:145"><i><font color="Green" title="E170">A1</font></i></a>, <a class="ref" href="jordan2c.html#T135" title="JORDAN2C:th.135">JORDAN2C:135</a></i>;<br/><a NAME="E6:145_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c25">x</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T45" title="TOPREAL1:th.45">TOPREAL1:45</a></i>;<br/><a NAME="E7:145_1_1"/><b>then </b>
<font color="Maroon" title="c23">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="txt" href="jordan.html#E4:145"><i><font color="Green" title="E173">A4</font></i></a>, <a class="txt" href="jordan.html#E5:145_1_1"><i><font color="Green" title="E178">A11</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E8:145_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E3:145"><i><font color="Green" title="E172">A3</font></i></a>, <a class="ref" href="jordan.html#T14" target="_self" title="JORDAN:th.14">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="145_1_2"><b>suppose </b></a><a NAME="E1:145_1_2"/><i><font color="Green" title="E177">A12</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c26">E</font> =  <a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <font color="Maroon" title="c25">x</font>;<br/><a NAME="E2:145_1_2"/>
 <a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <font color="Maroon" title="c25">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">e</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><a NAME="E1:145_1_2_1"/><i><font color="Green" title="E178">A13</font></i>: 
<font color="Maroon" title="c27">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <font color="Maroon" title="c25">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c28">e</font> = <font color="Maroon" title="c27">e</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> ;<br/>
<a NAME="E3:145_1_2_1"/><i><font color="Green" title="E179">A14</font></i>: 
( <font color="Maroon" title="c28">e</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c28">e</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="c25">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jordan.html#E1:145_1_2_1"><i><font color="Green" title="E178">A13</font></i></a>, <a class="ref" href="topreal1.html#D15" title="TOPREAL1:def.15">TOPREAL1:def 15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:145_1_2_1_1"/>
<font color="Maroon" title="c28">e</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3
 ;<br/><a NAME="E2:145_1_2_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">p</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> st <br/>( <font color="Maroon" title="c28">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 )
 ;<br/><b>hence </b><a NAME="E3:145_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E9:145"><i><font color="Green" title="E176">A7</font></i></a>, <a class="txt" href="jordan.html#E1:145_1_2"><i><font color="Green" title="E177">A12</font></i></a>, <a class="txt" href="jordan.html#E3:145_1_2_1"><i><font color="Green" title="E179">A14</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>
<b>hence </b><a NAME="E5:145_1_2_1"/>
<font color="Maroon" title="c27">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:145_1_2"/><b>then </b>
 <a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <font color="Maroon" title="c25">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E7:145"><i><font color="Green" title="E175">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:145_1_2"/><b>then </b>
 <a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <font color="Maroon" title="c25">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/><a NAME="E5:145_1_2"/><b>then </b><i><font color="Green" title="E178">A15</font></i>: 
 <a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <font color="Maroon" title="c25">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:145"><i><font color="Green" title="E170">A1</font></i></a>, <a class="ref" href="jordan2c.html#T134" title="JORDAN2C:th.134">JORDAN2C:134</a></i>;<br/><a NAME="E6:145_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <font color="Maroon" title="c25">x</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T45" title="TOPREAL1:th.45">TOPREAL1:45</a></i>;<br/><a NAME="E7:145_1_2"/><b>then </b>
<font color="Maroon" title="c23">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c22">P</font>
 <b>by </b><i><a class="txt" href="jordan.html#E4:145"><i><font color="Green" title="E173">A4</font></i></a>, <a class="txt" href="jordan.html#E5:145_1_2"><i><font color="Green" title="E178">A15</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E8:145_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E3:145"><i><font color="Green" title="E172">A3</font></i></a>, <a class="ref" href="jordan.html#T14" target="_self" title="JORDAN:th.14">Th14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
