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<b>let </b><font color="Maroon">S</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E72</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> #)
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:10_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">B</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="E2:10_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<br/><b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P</font>, <font color="Maroon">Q1</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E4:10_1"/>
<font color="Maroon">P1</font>,<font color="Maroon">Q1</font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E5:10_1"/><b>then </b>
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font> or <font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font> )
 <b>by </b><i>, , , <a class="ref" href="connsp_1.html#T16">CONNSP_1:16</a></i>;<br/><b>hence </b><a NAME="E6:10_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> or <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
<font color="Maroon">B</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T16">CONNSP_1:16</a></i>;<br/>


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