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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>thus </b><a NAME="E2:32"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V2" target="_self">everywhere_dense</a> implies ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1">open</a> <a href="tex_3.html#V1" target="_self">dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V2" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E2:32_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i><a class="ref" href="tex_3.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:32_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:32_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:32_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T41">TOPS_3:41</a></i>;<br/>
<a NAME="E7:32_1"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="tex_3.html#E6:32_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tops_3.html#T17">TOPS_3:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1">open</a> <a href="tex_3.html#V1" target="_self">dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:32_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E5:32_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="tex_3.html#E6:32_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tex_3.html#T23" target="_self">Th23</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E10:32_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E4:32_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="tex_3.html#E9:32_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>4</sub></font> being  <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1">open</a> <a href="tex_3.html#V1" target="_self">dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E3:32"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E1:32_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="tops_1.html#V1">dense</a> )
 <b>by </b><i><a class="txt" href="tex_3.html#E3:32"><i><font color="Green">E15</font></i></a>, <a class="ref" href="tex_3.html#D1" target="_self">Def1</a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a>, <a class="ref" href="tsep_1.html#T16">TSEP_1:16</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:32"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T41">TOPS_3:41</a></i>;<br/>
<b>hence </b><a NAME="E7:32"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V2" target="_self">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tex_3.html#D2" target="_self">Def2</a></i>;<br/>


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