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

<b>let </b><font color="Maroon">S</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="E1:56"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S</font> <a href="jordan1.html#NR1" target="_self">has_property_J</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">S'</font> = <font color="Maroon">S</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/>;<br/>
<b>consider </b><font color="Maroon">A1</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>, <font color="Maroon">A2</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="E4:56"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>and </b><br/><a NAME="E5:56"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font>
 <b>and </b><br/><a NAME="E6:56"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font>
 <b>and </b><br/><a NAME="E7:56"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">C1</font>, <font color="Olive">C2</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>  st <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> holds <br/>( <font color="Olive">C1</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">C2</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:56"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:56"/><i><font color="Green">E37</font></i>: 
 <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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E10:56"/>
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G0A</font> = <font color="Maroon">A1</font>, <font color="Maroon">G0B</font> = <font color="Maroon">A2</font> as   <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> <font color="Maroon">S'</font></span>)</span> ;<br/>
<a NAME="E12:56"/>
( <font color="Maroon">G0A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">G0B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 
;<br/>
<a NAME="E13:56"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">G0A</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> <font color="Maroon">S'</font> &amp; <font color="Maroon">G0B</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> <font color="Maroon">S'</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C3</font> be   <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> <font color="Maroon">S'</font></span>)</span>;<br/><b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">C3</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> <font color="Maroon">S'</font>
 ;<br/><a NAME="E2:56_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">C3</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> <font color="Maroon">S'</font></span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#T34">CONNSP_1:34</a></i>;<br/><a NAME="E3:56_1"/>
<font color="Maroon">C3</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G0A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G0B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E4:56_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">C3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G0A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G0B</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="ref" href="jordan1.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1"/>
( <font color="Maroon">C3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G0A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i><a class="ref" href="jordan1.html#T8" target="_self">Th8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1"/>
<font color="Maroon">C3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G0A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:56_1_1_1_1"/><b>then </b>
<font color="Maroon">C3</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G0A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:56_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
not <font color="Maroon">C3</font>,<font color="Maroon">G0A</font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="connsp_1.html#T2">CONNSP_1:2</a></i>;<br/><a NAME="E4:56_1_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">C3</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">G0A</font> is <a href="connsp_1.html#V2">connected</a> )
 <b>by </b><i><a class="ref" href="jordan1.html#T6" target="_self">Th6</a>, , <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/><a NAME="E5:56_1_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">C3</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G0A</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T18">CONNSP_1:18</a></i>;<br/><a NAME="E6:56_1_1_1_1"/>
<font color="Maroon">G0A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C3</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G0A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:56_1_1_1_1"/><b>then </b>
<font color="Maroon">G0A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G0A</font>
 <b>by </b><i><a class="ref" href="jordan1.html#T6" target="_self">Th6</a>, , <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/><a NAME="E8:56_1_1_1_1"/><b>then </b>
<font color="Maroon">C3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G0A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E9:56_1_1_1_1"/>
( <font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G0A</font> or <font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G0B</font> )
 <b>by </b><i>, , <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_2"/>
<font color="Maroon">C3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:56_1_1_1_2"/><b>then </b>
<font color="Maroon">C3</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G0B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:56_1_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
not <font color="Maroon">C3</font>,<font color="Maroon">G0B</font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="connsp_1.html#T2">CONNSP_1:2</a></i>;<br/><a NAME="E4:56_1_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">C3</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">G0B</font> is <a href="connsp_1.html#V2">connected</a> )
 <b>by </b><i><a class="ref" href="jordan1.html#T6" target="_self">Th6</a>, , <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/><a NAME="E5:56_1_1_1_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">C3</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G0B</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T18">CONNSP_1:18</a></i>;<br/><a NAME="E6:56_1_1_1_2"/>
<font color="Maroon">G0B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C3</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G0B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:56_1_1_1_2"/><b>then </b>
<font color="Maroon">G0B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C3</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">G0B</font>
 <b>by </b><i><a class="ref" href="jordan1.html#T6" target="_self">Th6</a>, , <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/><a NAME="E8:56_1_1_1_2"/><b>then </b>
<font color="Maroon">C3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G0B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E9:56_1_1_1_2"/>
( <font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G0A</font> or <font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G0B</font> )
 <b>by </b><i>, , <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:56_1"/>
( <font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G0A</font> or <font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G0B</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:56"/>
( <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  ex <font color="Olive">A1</font>, <font color="Olive">A2</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> ex <font color="Olive">C1</font>, <font color="Olive">C2</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> st <br/>( <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Olive">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">A2</font> &amp; <font color="Olive">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Olive">A2</font> &amp; <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Olive">A1</font> &amp; <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> &amp; <font color="Olive">C1</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">C2</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; ( for <font color="Olive">C3</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">C3</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">S</font> <a href="subset_1.html#K3">`</a> </span>)</span> or <font color="Olive">C3</font> <a href="hidden.html#R1">=</a> <font color="Olive">C1</font> or <font color="Olive">C3</font> <a href="hidden.html#R1">=</a> <font color="Olive">C2</font> ) ) ) )
 <b>by </b><i>, , , <a class="ref" href="jordan1.html#T6" target="_self">Th6</a></i>;<br/>


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