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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><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>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">C</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/>
<a NAME="E3:24"/>
( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c5">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_3.html#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tex_3.html#T16" target="_self" title="TEX_3:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E4:24"/>
( <font color="Maroon" title="c4">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> iff <font color="Maroon" title="c3">A</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_3.html#E1:24"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="tops_3.html#T64" title="TOPS_3:th.64">TOPS_3:64</a></i>;<br/>


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