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<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X'</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fintopo6.html#M1" target="_self">SubSpace</a> of <font color="Maroon">FT</font>;<br/><b>let </b><font color="Maroon">A8</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/><b>let </b><font color="Maroon">B8</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font>;<br/>







<b>assume </b><a NAME="E1:49"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a> &amp; <font color="Maroon">A8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B8</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
( <font color="Maroon">A8</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A8</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:49_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">A8</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font>
 ;<br/><a NAME="E3:49_1_1"/>
<font color="Maroon">B8</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X'</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E4:49_1_1"/>
( <font color="Maroon">A8</font> is <a href="fin_topo.html#V5">connected</a> iff <font color="Maroon">B8</font> is <a href="fin_topo.html#V5">connected</a> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">A8</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A8</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font> ;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B8</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font> <b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, </i>;<br/><b>reconsider </b><font color="Maroon">FT'</font> = <font color="Maroon">FT</font>, <font color="Maroon">X</font> = <font color="Maroon">X'</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/><b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT'</font> ;<br/><b>reconsider </b><font color="Maroon">B'</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font> ;<br/><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_1"/>
not <font color="Maroon">A8</font> is <a href="fin_topo.html#V5">connected</a>
 ;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E3:49_1_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">A8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E4:49_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E5:49_1_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E6:49_1_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E7:49_1_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/><a NAME="E8:49_1_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A8</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="fintopo6.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="fintopo6.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E10:49_1_2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A8</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="fintopo6.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="fintopo6.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E12:49_1_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">P'</font> <a href="fin_topo.html#K9">^b</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:49_1_2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">Q'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/><span class="p1">(<span class="default"><font color="Maroon">P'</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="fintopo6.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/><a NAME="E15:49_1_2_1"/><b>then </b>
<font color="Maroon">P'</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E16:49_1_2_1"/>
not <font color="Maroon">B8</font> is <a href="fin_topo.html#V5">connected</a>
 <b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a>, , , , <a class="ref" href="fintopo6.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_2"/>
not <font color="Maroon">B</font> is <a href="fin_topo.html#V5">connected</a>
 ;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font><b> such that </b><br/><a NAME="E3:49_1_2_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">B8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E4:49_1_2_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E5:49_1_2_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E6:49_1_2_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E7:49_1_2_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/><a NAME="E8:49_1_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:49_1_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:49_1_2_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P'</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">P'</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">P'</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="fintopo6.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="fintopo6.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/><a NAME="E14:49_1_2_2"/><b>then </b>
<font color="Maroon">P'</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E15:49_1_2_2"/>
not <font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a>
 <b>by </b><i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a>, <a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="fintopo6.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:49_1_2"/>
( <font color="Maroon">A8</font> is <a href="fin_topo.html#V5">connected</a> iff <font color="Maroon">B8</font> is <a href="fin_topo.html#V5">connected</a> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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