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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c3">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">X1</font> #)
 ;<br/>

<b>thus </b><a NAME="E2:28"/>
( <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">Y2</font> = <font color="Maroon" title="c3">X2</font> as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="tex_3.html#E1:28"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tmap_1.html#T12" title="TMAP_1:th.12">TMAP_1:12</a></i>;<br/>
<a NAME="E3:28_1"/>
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="c4">Y2</font> holds <br/><font color="Olive" title="b1">A2</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tex_3.html#E1:28"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="tex_3.html#E1:28_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">Def2</a></i>;<br/>


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

<b>assume </b><a NAME="E3:28"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">Y1</font> = <font color="Maroon" title="c2">X1</font> as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="tex_3.html#E1:28"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tmap_1.html#T12" title="TMAP_1:th.12">TMAP_1:12</a></i>;<br/>
<a NAME="E5:28"/>
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="c4">Y1</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>
 
<b>by </b><i><a class="txt" href="tex_3.html#E1:28"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="tex_3.html#E3:28"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:28"/>
<font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">Def2</a></i>;<br/>


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