<?xml version="1.0"?>
<div class="add">

<a NAME="E1:12_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</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>consider </b><font color="Maroon">c<sub>3</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</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="E4:12_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T10">TSEP_1:10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:12_1_1_1"/>
<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>3</sub></font>
 ;<br/><a NAME="E2:12_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E4:12_1_1"><i><font color="Green">E10</font></i></a></i>;<br/><b>hence </b><a NAME="E3:12_1_1_1"/>
<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#T16">TOPS_3:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:12_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="tex_3.html#V1" target="_self">dense</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V1">strict</a> &amp; not <font color="Maroon">c<sub>3</sub></font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i><a class="ref" href="tex_3.html#D1" target="_self">Def1</a></i>;<br/>


</div>
