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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 ;<br/>

<b>thus </b><a NAME="E2:11"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>, <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<a NAME="E1:11_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T2">TDLAT_3:2</a></i>;<br/>
<a NAME="E2:11_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="tsep_2.html#E1:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tsep_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>, <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 <b>by </b><i><a class="ref" href="tsep_2.html#T5" target="_self">Th5</a></i>;<br/>


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

<b>thus </b><a NAME="E3:11"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font>, <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<a NAME="E1:11_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tops_1.html#D1">TOPS_1:def 1</a></i>;<br/>
<a NAME="E2:11_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="tsep_2.html#E1:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tsep_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:11_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font>, <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 <b>by </b><i><a class="ref" href="tsep_2.html#T5" target="_self">Th5</a></i>;<br/>


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


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