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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="E1:55_3_1"/><i><font color="Green">E17</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="borsuk_1.html#V1">closed</a>
 ;<br/>

<b>assume </b><a NAME="E2:55_3_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V2" target="_self">proper</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</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="E4:55_3_1"/><i><font color="Green">E18</font></i>: 
<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>2</sub></font>
 <b>and </b><br/><a NAME="E5:55_3_1"/><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V1" target="_self">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E6:55_3_1"/><i><font color="Green">E20</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="tex_2.html#E5:55_3_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tex_2.html#T5" target="_self">Th5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_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="E1:55_3_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>2</sub></font>
 ;<br/><a NAME="E2:55_3_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_2.html#E4:55_3_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="tex_2.html#E6:55_3_1"><i><font color="Green">E20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:55_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_2.html#E1:55_3_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="borsuk_1.html#D14">BORSUK_1:def 14</a></i>;<br/>


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