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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">Y</font>, <font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</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="E1:87"/><i><font color="Green" title="E56">A1</font></i>: 
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">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="c6">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">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="c7">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</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>assume </b><a NAME="E5:87"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 ;<br/>

<b>thus </b><a NAME="E6:87"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<a NAME="E1:87_1"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c5">A1</font>,<font color="Maroon" title="c6">A2</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#E5:87"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="tsep_1.html#D8" target="_self" title="TSEP_1:def.8">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">D1</font>, <font color="Maroon" title="c9">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="E1:87_1_1"/>
( <font color="Maroon" title="c8">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#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font></span>)</span> &amp; <font color="Maroon" title="c9">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="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font></span>)</span> )
 ;<br/><a NAME="E2:87_1_1"/><b>then </b>
( <font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D1</font> &amp; <font color="Maroon" title="c6">A2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">D2</font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:87"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="tsep_1.html#D5" target="_self" title="TSEP_1:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E3:87_1_1"/>
<font color="Maroon" title="c8">D1</font>,<font color="Maroon" title="c9">D2</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#E1:87_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="tsep_1.html#T43" target="_self" title="TSEP_1:th.43">Th43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:87_1"/>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D8" target="_self" title="TSEP_1:def.8">Def8</a></i>;<br/>


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

<b>thus </b><a NAME="E7:87"/>
<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<a NAME="E1:87_2"/>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 
<b>by </b><i><a class="txt" href="tsep_1.html#E1:87"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="tsep_1.html#E6:87"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="tsep_1.html#T29" target="_self" title="TSEP_1:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E2:87_2"/>
<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:87"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="tsep_1.html#T29" target="_self" title="TSEP_1:th.29">Th29</a></i>;<br/>


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


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