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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>







<b>thus </b><a NAME="E1:53"/>
( ( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  or <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  ) implies <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  or <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )
 ;<br/>

<div><i><font color="Green" title="E33">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ;<br/><a NAME="E2:53_1_1"/>
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1"/>
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:53_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="connsp_1.html#T8" title="CONNSP_1:th.8">CONNSP_1:8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1_2"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ;<br/><a NAME="E2:53_1_2"/>
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E3:53_1_2"/>
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:53_1_2"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="connsp_1.html#T8" title="CONNSP_1:th.8">CONNSP_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:53_1"/>
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font>,<font color="Maroon" title="c4">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:53_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="tsep_1.html#E2:53_1"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>


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