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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_3.html#V3" title="TEX_3:attr.3">boundary</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>assume </b><a NAME="E1:39_4_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</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/>

<b>assume </b><a NAME="E2:39_4_1"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</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/>
<a NAME="E4:39_4_1"/>
<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#E2:39_4_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">Def2</a></i>;<br/>
<a NAME="E5:39_4_1"/><b>then </b>
( not <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> )
 
<b>by </b><i><a class="txt" href="tex_3.html#E1:39_4_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="tex_3.html#D3" target="_self" title="TEX_3:def.3">Def3</a>, <a class="ref" href="tops_3.html#T37" title="TOPS_3:th.37">TOPS_3:37</a></i>;<br/>
<b>hence </b><a NAME="E6:39_4_1"/>
contradiction
 ; <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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