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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> (  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  )</span><br/>



<b>thus </b><a NAME="E1:12"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1"/>
<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 
<b>by </b><i><a class="ref" href="tsep_2.html#T6" target="_self" title="TSEP_2:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E2:12_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 <b>by </b><i><a class="ref" href="tsep_2.html#T10" target="_self" title="TSEP_2:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:12"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<a NAME="E1:12_2"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ,<font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 
<b>by </b><i><a class="ref" href="tsep_2.html#T6" target="_self" title="TSEP_2:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E2:12_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 <b>by </b><i><a class="ref" href="tsep_2.html#T10" target="_self" title="TSEP_2:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>hence </b><a NAME="E3:12"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> </span><br/>

<b>thus </b><a NAME="E4:12"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 <b>by </b><i><a class="txt" href="tsep_2.html#E1:12"><i><font color="Green" title="E9">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
