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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> implies  not <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a> )</span><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/>
<b>assume </b><a NAME="E2:71_2_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a></span><br/>

<a NAME="E3:71_2_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">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/>
<a NAME="E4:71_2_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</a></i>;<br/>
<a NAME="E5:71_2_1"/><b>then </b>
not <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="ref" href="tex_1.html#T36" title="TEX_1:th.36">TEX_1:36</a></i>;<br/>
<b>hence </b><a NAME="E6:71_2_1"/>
not <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tex_3.html#T16" target="_self" title="TEX_3:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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