<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c2">A0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:80_2_1"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c2">A0</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>and </b><br/><a NAME="E3:80_2_1"/><i><font color="Green" title="E29">A8</font></i>: 
( <font color="Maroon" title="c2">A0</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c2">A0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="tex_1.html#T38" title="TEX_1:th.38">TEX_1:38</a></i>;<br/>
<a NAME="E4:80_2_1"/>
not <font color="Maroon" title="c2">A0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="#E3:80_2_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="tops_3.html#T17" title="TOPS_3:th.17">TOPS_3:17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">X0</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:80_2_1"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c2">A0</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="c3">X0</font>
 <b>by </b><i><a class="txt" href="#E3:80_2_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="tex_3.html#T23" target="_self" title="TEX_3:th.23">Th23</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">X0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> &amp; <font color="Maroon" title="c3">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> &amp; <font color="Maroon" title="c3">X0</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> &amp; <font color="Maroon" title="c3">X0</font> is <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> &amp; not <font color="Maroon" title="c3">X0</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )</span><br/>

<a NAME="E7:80_2_1"/>
<font color="Maroon" title="c2">A0</font> is <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>
 
<b>by </b><i><a class="txt" href="#E2:80_2_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:80_2_1"/>
( <font color="Maroon" title="c3">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> &amp; <font color="Maroon" title="c3">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> &amp; <font color="Maroon" title="c3">X0</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> &amp; <font color="Maroon" title="c3">X0</font> is <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> &amp; not <font color="Maroon" title="c3">X0</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 <b>by </b><i><a class="txt" href="#E6:80_2_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="tex_2.html#T13" title="TEX_2:th.13">TEX_2:13</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>
