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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">X2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font> #)
 ;<br/>

<b>thus </b><a NAME="E2:28"/>
( <font color="Maroon">X1</font> is  <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> implies <font color="Maroon">X2</font> is  <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">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">E24</font></i>: 
<font color="Maroon">X1</font> is  <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Y2</font> = <font color="Maroon">X2</font> as    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="tmap_1.html#T12">TMAP_1:12</a></i>;<br/>
<a NAME="E3:28_1"/>
for <font color="Olive">A2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y2</font> holds <br/><font color="Olive">A2</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i>, <a class="ref" href="tex_3.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<font color="Maroon">X2</font> is  <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:28"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X2</font> is  <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">X1</font> as    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="tmap_1.html#T12">TMAP_1:12</a></i>;<br/>
<a NAME="E5:28"/>
for <font color="Olive">A1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y1</font> holds <br/><font color="Olive">A1</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i>, <a class="ref" href="tex_3.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E6:28"/>
<font color="Maroon">X1</font> is  <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div>
