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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V2" title="TEX_3:attr.2">everywhere_dense</a>  iff <font color="Olive" title="b2">X2</font> is  <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>  )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V2" title="TEX_3:attr.2">everywhere_dense</a>  iff <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>  ) )</span><br/>



<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:48"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  iff <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:48"/>
( <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  implies <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>  implies <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
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_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">TEX_3:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V3" title="TOPS_1:attr.3">nowhere_dense</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:48_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">A2</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> </span><br/><a NAME="E2:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/><a NAME="E3:48_1_1"/>
<font color="Maroon" title="c4">A1</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1"/>
<font color="Maroon" title="c6">A2</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="tex_3.html#T4" target="_self" title="TEX_3:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:48_1"/>
<font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_3.html#D4" target="_self" title="TEX_3:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:48"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
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#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tex_3.html#D4" target="_self" title="TEX_3:def.4">TEX_3:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:48_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">A1</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/><a NAME="E2:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/><a NAME="E3:48_2"/>
<font color="Maroon" title="c5">A2</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_2"/>
<font color="Maroon" title="c6">A1</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="tex_3.html#T4" target="_self" title="TEX_3:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:48"/>
<font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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