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

<b>let </b><font color="Maroon">c<sub>1</sub></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><a NAME="E1:125"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="jordan2c.html#V1" target="_self">Bounded</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="jordan1.html#V2">Jordan</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:125"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="jordan1.html#D2">JORDAN1:def 2</a></i>;<br/>
<i><font color="Green">E116</font></i>:  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/>
<i><font color="Green">E117</font></i>:  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/>
<a NAME="E6:125"/><b>then </b><i><font color="Green">E118</font></i>: 
(  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp;  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan2c.html#E4:125"><i><font color="Green">E116</font></i></a></i>;<br/>
<a NAME="E7:125"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:125"><i><font color="Green">E116</font></i></a>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E8:125"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:125"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan2c.html#E6:125"><i><font color="Green">E118</font></i></a>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:125"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jordan1.html#D2">JORDAN1:def 2</a></i>;<br/>
<a NAME="E10:125"/><i><font color="Green">E121</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font> is <a href="struct_0.html#V3">empty</a>
 
;<br/>
<a NAME="E11:125"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E6:125"><i><font color="Green">E118</font></i></a>, <a class="ref" href="connsp_1.html#T34">CONNSP_1:34</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:125"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:125"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jordan2c.html#T76" target="_self">Th76</a></i>;<br/>
<a NAME="E14:125"/>
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E13:125"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan2c.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E16:125"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_3"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E4:125"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan2c.html#E6:125"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan2c.html#E16:125"><i><font color="Green">E124</font></i></a>, <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_3_1"><b>suppose </b></a><a NAME="E1:125_3_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:125_3_1_1"/>
not <font color="Maroon">c<sub>3</sub></font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/><a NAME="E2:125_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:125"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jordan2c.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:125_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:125_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:125"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan2c.html#E11:125"><i><font color="Green">E122</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E5:125_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan2c.html#E16:125"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan2c.html#E1:125_3_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><a NAME="E3:125_3_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:125"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:125_3_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:125_3_1_2"/>
not  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:125_3_1_2"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:125_3_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:125_3_1_2"><i><font color="Green">E128</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:125_3_1_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:125_3_1_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:125_3_1_2"><i><font color="Green">E129</font></i></a></i>;<br/><a NAME="E9:125_3_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:125_3_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self">Def3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:125_3_1_2"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/><a NAME="E12:125_3_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E10:125"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan2c.html#E11:125_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="connsp_1.html#T34">CONNSP_1:34</a></i>;<br/><a NAME="E13:125_3_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> or <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:125"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan2c.html#E11:125_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/><a NAME="E14:125_3_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> or <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:125_3_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><a NAME="E2:125_3_1_2_1"/><b>then </b>
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:125"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan2c.html#E16:125"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan2c.html#E1:125_3_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan2c.html#E8:125_3_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan2c.html#E11:125_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E3:125_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E13:125"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan2c.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><a NAME="E16:125_3_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:125"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan2c.html#E5:125"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan2c.html#E6:125"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan2c.html#E3:125_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan2c.html#E6:125_3_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan2c.html#E8:125_3_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan2c.html#E11:125_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/><a NAME="E17:125_3_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:125_3_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="sprect_1.html#T7">SPRECT_1:7</a></i>;<br/><b>then </b><font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:125"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan2c.html#E6:125_3_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan2c.html#E8:125_3_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan2c.html#E11:125_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan2c.html#E14:125_3_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan2c.html#E16:125_3_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/><b>hence </b><a NAME="E19:125_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E12:125_3_1_2"><i><font color="Green">E133</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:125_3_1"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:125_3_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan2c.html#E4:125_3_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_3_2"><b>suppose </b></a><a NAME="E1:125_3_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><div class="add"><a NAME="E2:125_3_2"/><i><font color="Green">E126</font></i>: 
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E3:125_3_2"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self">Th31</a></i>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>;<br/><a NAME="E4:125_3_2"/>
<span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan2c.html#E4:125"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan2c.html#E5:125"><i><font color="Green">E117</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E5:125_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E6:125_3_2"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:125"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan2c.html#E6:125"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan2c.html#E10:125"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan2c.html#E16:125"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan2c.html#E1:125_3_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="jordan2c.html#T115" target="_self">Th115</a></i>;<br/><a NAME="E7:125_3_2"/><i><font color="Green">E129</font></i>: 
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:125_3_2_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:125_3_2_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:125_3_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:125_3_2_1"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E2:125_3_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan2c.html#E1:125_3_2_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E4:125_3_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan2c.html#E6:125_3_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan2c.html#E2:125_3_2_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:125_3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:125_3_2_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:125_3_2_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:125_3_2_2"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:125"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan2c.html#E6:125_3_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan2c.html#E1:125_3_2_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E4:125_3_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:125_3_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan2c.html#E2:125_3_2_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:125_3_2"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:125_3_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E10:125_3_2"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:125"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jordan2c.html#T114" target="_self">Th114</a></i>;<br/><b>hence </b><a NAME="E11:125_3_2"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">c<sub>1</sub></font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:125"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan2c.html#E9:125_3_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
