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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <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/>





<a NAME="E1:21"/><i><font color="Green">E12</font></i>: 
( <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">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> &amp; <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">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/>
<b>thus </b><a NAME="E2:21"/>
( <font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2">are_weakly_separated</a>  implies <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">A1</font>,<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">A2</font> <a href="connsp_1.html#R1">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font>,<font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="connsp_1.html#R1">are_separated</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tsep_1.html#D7">TSEP_1:def 7</a><br/>

<b>hence </b><a NAME="E2:21_1"/>
<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">A1</font>,<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">A2</font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:21"/>
<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">A1</font>,<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">A2</font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<br/>

<b>hence </b><a NAME="E4:21"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2">are_weakly_separated</a> 
 <b>by </b><i>, <a class="ref" href="tsep_1.html#D7">TSEP_1:def 7</a></i>;<br/>


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