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<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:80_2_1"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:80_2_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_1.html#T34" title="TEX_1:th.34">TEX_1:34</a></span>;<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_2_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="tops_3.html#T17" title="TOPS_3:th.17">TOPS_3:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:80_2_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_2_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="tex_3.html#T23" target="_self" title="TEX_3:th.23">Th23</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">X0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_2_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="subset_1.html#D6" title="SUBSET_1:def.6">SUBSET_1:def 6</a></span>;<br/>
<span class="kw">hence </span><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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:80_2_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="tex_2.html#T8" title="TEX_2:th.8">TEX_2:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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