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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>;<br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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>reconsider </b><font color="Maroon" title="c4">A1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E3:56"/>
( ( <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> &amp; <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> ) or ( <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> &amp; <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> ) )
 ;<br/>

<a NAME="E4:56"/><b>then </b>
( ( <font color="Maroon" title="c4">A1</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> &amp; <font color="Maroon" title="c5">A2</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> ) or ( <font color="Maroon" title="c4">A1</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Maroon" title="c5">A2</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> ) )
 
<b>by </b><i><a class="ref" href="tex_3.html#D3" target="_self" title="TEX_3:def.3">Def3</a>, <a class="ref" href="tex_3.html#D4" target="_self" title="TEX_3:def.4">Def4</a></i>;<br/>
<a NAME="E5:56"/><b>then </b>
<font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">A2</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T30" title="TOPS_3:th.30">TOPS_3:30</a></i>;<br/>
<a NAME="E6:56"/><b>then </b>
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>  st <font color="Olive" title="b1">A</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">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font></span>)</span> holds <br/><font color="Olive" title="b1">A</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:56"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a>
 <b>by </b><i><a class="ref" href="tex_3.html#D3" target="_self" title="TEX_3:def.3">Def3</a></i>;<br/>


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