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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  implies <font color="Maroon" title="c2">X0</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( <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="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X0</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</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="c2">X0</font> implies <font color="Maroon" title="c3">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:29_1_1_1"/>
<font color="Maroon" title="c3">A</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="c2">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/><a NAME="E2:29_1_1_1"/><span class="kw">then </span>
( <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="c3">A</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="txt" href="#E1:29_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></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></span>;<br/><span class="kw">hence </span><a NAME="E3:29_1_1_1"/>
<font color="Maroon" title="c3">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T36" title="TOPS_3:th.36">TOPS_3:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:29_1_1"/>
<font color="Maroon" title="c2">X0</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_3.html#D2" target="_self" title="TEX_3:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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