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<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>assume </b><a NAME="E1:11_3_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V2">proper</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="borsuk_1.html#V1">closed</a> )
 ;<br/>

<b>assume </b><a NAME="E2:11_3_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V1" target="_self">dense</a>
 ;<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/>
<a NAME="E4:11_3_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="txt" href="tex_3.html#E1:11_3_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="tex_3.html#E2:11_3_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="tex_3.html#T9" target="_self">Th9</a>, <a class="ref" href="tsep_1.html#T11">TSEP_1:11</a></i>;<br/>
<a NAME="E5:11_3_1"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a>, <a class="ref" href="tops_3.html#D2">TOPS_3:def 2</a></i>;<br/>
<a NAME="E6:11_3_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V1">proper</a>
 
<b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>
<b>hence </b><a NAME="E7:11_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_3.html#E1:11_3_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tex_2.html#T13">TEX_2:13</a></i>;<br/>


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