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

<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>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:146"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a>
 <b>and </b><br/><a NAME="E2:146"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="jordan24.html#R1">realize-max-dist-in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:146"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">A</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">P</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="E4:146"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:146"/><i><font color="Green">E79</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> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 ;<br/>

<a NAME="E6:146"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="jordan.html#T51" target="_self">Th51</a>, </i>;<br/>
<a NAME="E7:146"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<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/>;<br/>
<a NAME="E9:146"/><i><font color="Green">E81</font></i>: 
(  not  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 or  not  <a href="real_1.html#K1">-</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 
<b>by </b><i><a class="ref" href="jordan.html#T57" target="_self">Th57</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="146_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:146_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="146_1_1"><b>suppose </b></a><a NAME="E1:146_1_1"/><i><font color="Green">E82</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">E</font> =  <a href="topreal1.html#K7">east_halfline</a> <font color="Maroon">x</font>;<br/><a NAME="E2:146_1_1"/>
 <a href="topreal1.html#K7">east_halfline</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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><a NAME="E1:146_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K7">east_halfline</a> <font color="Maroon">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</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> ;<br/>
<a NAME="E3:146_1_1_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">e</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">e</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D13">TOPREAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:146_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 ;<br/><a NAME="E2:146_1_1_1_1"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 ;<br/><b>hence </b><a NAME="E3:146_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:146_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:146_1_1"/><b>then </b>
 <a href="topreal1.html#K7">east_halfline</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:146_1_1"/><b>then </b>
 <a href="topreal1.html#K7">east_halfline</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E5:146_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="topreal1.html#K7">east_halfline</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan.html#T50" target="_self">Th50</a>, <a class="ref" href="jordan2c.html#T135">JORDAN2C:135</a></i>;<br/><a NAME="E6:146_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K7">east_halfline</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T45">TOPREAL1:45</a></i>;<br/><a NAME="E7:146_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E8:146_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T52" target="_self">Th52</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="146_1_2"><b>suppose </b></a><a NAME="E1:146_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">E</font> =  <a href="topreal1.html#K9">west_halfline</a> <font color="Maroon">x</font>;<br/><a NAME="E2:146_1_2"/>
 <a href="topreal1.html#K9">west_halfline</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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><a NAME="E1:146_1_2_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K9">west_halfline</a> <font color="Maroon">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</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> ;<br/>
<a NAME="E3:146_1_2_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">e</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">e</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D15">TOPREAL1:def 15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:146_1_2_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 ;<br/><a NAME="E2:146_1_2_1_1"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 3 )
 ;<br/><b>hence </b><a NAME="E3:146_1_2_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:146_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3</span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:146_1_2"/><b>then </b>
 <a href="topreal1.html#K9">west_halfline</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:146_1_2"/><b>then </b>
 <a href="topreal1.html#K9">west_halfline</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E5:146_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="topreal1.html#K9">west_halfline</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan.html#T50" target="_self">Th50</a>, <a class="ref" href="jordan2c.html#T134">JORDAN2C:134</a></i>;<br/><a NAME="E6:146_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K9">west_halfline</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T45">TOPREAL1:45</a></i>;<br/><a NAME="E7:146_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E8:146_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T52" target="_self">Th52</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
