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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#V1" title="TEX_3:attr.1">dense</a> &amp; <font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</a> implies  not <font color="Maroon" title="c2">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</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>reconsider </b><font color="Maroon" title="c4">B</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="E3:73_6_1"/><i><font color="Green" title="E27">A6</font></i>: 
( <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> &amp; <font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">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="tsep_1.html#V1" title="TSEP_1:attr.1">open</a></span><br/>

<a NAME="E4:73_6_1"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
( <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c4">B</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> )
 
<b>by </b><i><a class="ref" href="tex_3.html#T9" target="_self" title="TEX_3:th.9">Th9</a>, <a class="ref" href="tex_2.html#T13" title="TEX_2:th.13">TEX_2:13</a></i>;<br/>
<b>assume </b><a NAME="E5:73_6_1"/><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c2">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_6_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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> &amp; <font color="Maroon" title="c5">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/><b>thus </b><a NAME="E1:73_6_1_1"/>
<font color="Maroon" title="c5">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="txt" href="tex_3.html#E4:73_6_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="tex_2.html#T5" title="TEX_2:th.5">TEX_2:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/><b>thus </b><a NAME="E2:73_6_1_1"/>
( <font color="Maroon" title="c5">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="tex_3.html#E3:73_6_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="tex_3.html#E5:73_6_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="tex_3.html#T9" target="_self" title="TEX_3:th.9">Th9</a>, <a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:73_6_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tex_1.html#T38" title="TEX_1:th.38">TEX_1:38</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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