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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">X1</font>, <font color="Olive" title="b2">X2</font> being   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">X2</font> <a href="tsep_2.html#R3" title="TSEP_2:pred.3">constitute_a_decomposition</a>  holds <br/>( <font color="Olive" title="b1">X1</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> iff <font color="Olive" title="b2">X2</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> )</span><br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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"> ( <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font> <a href="tsep_2.html#R3" title="TSEP_2:pred.3">constitute_a_decomposition</a>  implies ( <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> iff <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> ) )</span><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>, <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="E2:41"/><i><font color="Green" title="E19">A1</font></i>: 
for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</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 <font color="Maroon" title="c2">X1</font> &amp; <font color="Olive" title="b2">A2</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 <font color="Maroon" title="c3">X2</font> holds <br/><font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tsep_2.html#D2" title="TSEP_2:def.2">TSEP_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> iff <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> )</span><br/>

<b>thus </b><a NAME="E3:41"/>
( <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> implies <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a> implies <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E20">A2</font></i>: 
for <font color="Olive" title="b1">A1</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">A1</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 <font color="Maroon" title="c2">X1</font> holds <br/><font color="Olive" title="b1">A1</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tex_3.html#D1" target="_self" title="TEX_3:def.1">TEX_3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A2</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"> ( <font color="Maroon" title="c6">A2</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 <font color="Maroon" title="c3">X2</font> implies <font color="Maroon" title="c6">A2</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> )</span><br/><b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon" title="c6">A2</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 <font color="Maroon" title="c3">X2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A2</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a></span><br/><a NAME="E2:41_1_1"/><b>then </b>
( <font color="Maroon" title="c4">A1</font>,<font color="Maroon" title="c6">A2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c4">A1</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 <b>by </b><i><a class="txt" href="tex_3.html#E2:41"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="tex_3.html#E1:41_1"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:41_1_1"/>
<font color="Maroon" title="c6">A2</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 <b>by </b><i><a class="ref" href="tex_3.html#T2" target="_self" title="TEX_3:th.2">Th2</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="E3:41_1"/>
<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>; <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>assume </b><a NAME="E4:41"/><i><font color="Green" title="E20">A3</font></i>: 
for <font color="Olive" title="b1">A2</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">A2</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 <font color="Maroon" title="c3">X2</font> holds <br/><font color="Olive" title="b1">A2</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tex_3.html#D3" target="_self" title="TEX_3:def.3">TEX_3:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A1</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"> ( <font color="Maroon" title="c6">A1</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 <font color="Maroon" title="c2">X1</font> implies <font color="Maroon" title="c6">A1</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )</span><br/><b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon" title="c6">A1</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 <font color="Maroon" title="c2">X1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A1</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a></span><br/><a NAME="E2:41_2"/><b>then </b>
( <font color="Maroon" title="c6">A1</font>,<font color="Maroon" title="c5">A2</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a>  &amp; <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="txt" href="tex_3.html#E2:41"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="tex_3.html#E4:41"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:41_2"/>
<font color="Maroon" title="c6">A1</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tex_3.html#T2" target="_self" title="TEX_3:th.2">Th2</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="E6:41"/>
<font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tex_3.html#D1" target="_self" title="TEX_3:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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