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<b>let </b><font color="Maroon" title="c2">X0</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:11_1_1"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> &amp; <font color="Maroon" title="c2">X0</font> is <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<a NAME="E3:11_1_1"/>
( <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c3">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="tex_3.html#E1:11_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="tex_3.html#T9" target="_self" title="TEX_3:th.9">Th9</a>, <a class="ref" href="tsep_1.html#T11" title="TSEP_1:th.11">TSEP_1:11</a></i>;<br/>
<a NAME="E4:11_1_1"/><b>then </b>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</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="c1">X</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a>, <a class="ref" href="tops_3.html#D2" title="TOPS_3:def.2">TOPS_3:def 2</a></i>;<br/>
<a NAME="E5:11_1_1"/><b>then </b>
not <font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 
<b>by </b><i><a class="ref" href="tex_2.html#T5" title="TEX_2:th.5">TEX_2:5</a></i>;<br/>
<b>hence </b><a NAME="E6:11_1_1"/>
not <font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T13" title="TEX_2:th.13">TEX_2:13</a></i>;<br/>


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