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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Olive" title="b1">D</font></span>)</span>  st <font color="Olive" title="b1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b1">D</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> holds <br/><font color="Olive" title="b2">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a></span><br/><b>let </b><font color="Maroon" title="c2">D</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">D</font></span>)</span>  st <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c2">D</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> holds <br/><font color="Olive" title="b1">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a></span><br/><b>let </b><font color="Maroon" title="c3">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">D</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c2">D</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> implies <font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> )</span><br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 ; <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">D</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> or <font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> )</span><br/>

<b>assume </b><a NAME="E2:5"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">D</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">E</font> = <font color="Maroon" title="c2">D</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">D</font></span>)</span> <b>by </b><i><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></i>;<br/>
<a NAME="E4:5"/>
( <font color="Maroon" title="c4">E</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c4">E</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="tex_1.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="tex_1.html#T4" target="_self" title="TEX_1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:5"/>
<font color="Maroon" title="c3">C</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_1.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="tops_3.html#T36" title="TOPS_3:th.36">TOPS_3:36</a>, <a class="ref" href="tops_3.html#T38" title="TOPS_3:th.38">TOPS_3:38</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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