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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">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Y1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">Y2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><a NAME="E1:102"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">Y2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E6:102"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#D3" target="_self">Def3</a>, </i>;<br/>
<b>assume </b><a NAME="E7:102"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 ;<br/>

<a NAME="E8:102"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>thus </b><a NAME="E9:102"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">D2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:102_1_1"/>
( <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y1</font></span>)</span> &amp; <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X2</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font></span>)</span> )
 ;<br/><a NAME="E2:102_1_1"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:102_1_1"/>
<font color="Maroon">D1</font>,<font color="Maroon">D2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D5" target="_self">Def5</a>, <a class="ref" href="tsep_1.html#T29" target="_self">Th29</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:102_1"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">Y2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T16" target="_self">Th16</a></i>;<br/>


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

<b>hence </b><a NAME="E10:102"/>
<font color="Maroon">Y1</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X1</font>,<font color="Maroon">Y2</font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">X2</font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 ;<br/>


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