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<b>let </b><font color="Maroon">X0</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:21_3_1"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">X0</font> is <a href="tex_2.html#V2">proper</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:21_3_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/><a NAME="E2:21_3_1_1"/><b>then </b>
not <font color="Maroon">A</font> is <a href="tex_2.html#V1">proper</a>
 <b>by </b><i>, <a class="ref" href="tex_2.html#T13">TEX_2:13</a></i>;<br/><a NAME="E3:21_3_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/><a NAME="E4:21_3_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 ;<br/><b>hence </b><a NAME="E5:21_3_1_1"/>
<font color="Maroon">A</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T31">TOPS_3:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_3_1"/>
<font color="Maroon">X0</font> is <a href="tex_3.html#V2" target="_self">everywhere_dense</a>
 <b>by </b><i/>;<br/>


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