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<b>let </b><font color="Maroon">X</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>







<b>thus </b><a NAME="E1:68"/>
( <font color="Maroon">A1</font>,<font color="Maroon">B</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a>  &amp; <font color="Maroon">A2</font>,<font color="Maroon">B</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a>  implies <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>,<font color="Maroon">B</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/>
( <font color="Maroon">A1</font>,<font color="Maroon">B</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a>  &amp; <font color="Maroon">A2</font>,<font color="Maroon">B</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a>  )
 ;<br/>

<a NAME="E2:68_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>,<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="connsp_1.html#R1">are_separated</a>  &amp; <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>,<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> <a href="connsp_1.html#R1">are_separated</a>  )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:68_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="connsp_1.html#R1">are_separated</a>  &amp; <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="connsp_1.html#R1">are_separated</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E4:68_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:68_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>;<br/>
<a NAME="E6:68_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>,<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>;<br/>
<b>hence </b><a NAME="E7:68_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>,<font color="Maroon">B</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T5" target="_self">Th5</a></i>;<br/>


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


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