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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X2</font>, <font color="Maroon" title="c3">X1</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c5">Y2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>







<b>assume </b><a NAME="E1:102"/><i><font color="Green" title="E63">A1</font></i>: 
( <font color="Maroon" title="c4">Y1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X2</font> &amp; <font color="Maroon" title="c5">Y2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">A1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">B1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">B2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Y2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self" title="TSEP_1:th.1">Th1</a></i>;<br/>
<a NAME="E6:102"/><i><font color="Green" title="E64">A2</font></i>: 
( <font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A2</font> &amp; <font color="Maroon" title="c9">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A1</font> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E1:102"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="tsep_1.html#T4" target="_self" title="TSEP_1:th.4">Th4</a></i>;<br/>
<b>assume </b><a NAME="E7:102"/>
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> 
 ;<br/>

<a NAME="E8:102"/><b>then </b><i><font color="Green" title="E65">A3</font></i>: 
<font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c7">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#D9" target="_self" title="TSEP_1:def.9">Def9</a></i>;<br/>
<b>thus </b><a NAME="E9:102"/>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">Y1</font>,<font color="Maroon" title="c2">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">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" title="c10">D1</font>, <font color="Maroon" title="c11">D2</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>assume </b><a NAME="E1:102_1_1"/>
( <font color="Maroon" title="c10">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">Y1</font></span>)</span> &amp; <font color="Maroon" title="c11">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font></span>)</span> )
 ;<br/><a NAME="E2:102_1_1"/><b>then </b>
( <font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">D1</font> &amp; <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D2" target="_self" title="TSEP_1:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:102_1_1"/>
<font color="Maroon" title="c10">D1</font>,<font color="Maroon" title="c11">D2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E6:102"><i><font color="Green" title="E64">A2</font></i></a>, <a class="txt" href="tsep_1.html#E8:102"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="tsep_1.html#T56" target="_self" title="TSEP_1:th.56">Th56</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:102_1"/>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">Y1</font>,<font color="Maroon" title="c2">X2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D9" target="_self" title="TSEP_1:def.9">Def9</a></i>;<br/>


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

<b>hence </b><a NAME="E10:102"/>
<font color="Maroon" title="c4">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c5">Y2</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c2">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> 
 ;<br/>


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