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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  non <a href="struct_0.html#V3">empty</a> <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <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>4</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="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</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="E1:27_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</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/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/><a NAME="E3:27_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="tex_3.html#E1:27_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="tex_3.html#D2" target="_self">Def2</a>, <a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/><b>hence </b><a NAME="E4:27_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T64">TOPS_3:64</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:27"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <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="tex_3.html#D2" target="_self">Def2</a>, <a class="ref" href="tsep_1.html#T7">TSEP_1:7</a></i>;<br/>


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