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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="E1:21_2_1"/><i><font color="Green" title="E13">A2</font></i>: 
not <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>
 ;<br/>

<b>assume </b><a NAME="E2:21_2_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:21_2_1_1"/>
<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="c2">X0</font>
 ;<br/><a NAME="E2:21_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E2:21_2_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:21_2_1_1"/>
<font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T33" title="TOPS_3:th.33">TOPS_3:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_3.html#E1:21_2_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="tex_3.html#D1" target="_self" title="TEX_3:def.1">Def1</a></i>;<br/>


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