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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X0</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/>

<b>hence </b><a NAME="E2:20"/>
( <font color="Maroon">X0</font> is <a href="tex_3.html#V2" target="_self">everywhere_dense</a> implies <font color="Maroon">A</font> is <a href="tops_3.html#V1">everywhere_dense</a> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E3:20"/>
<font color="Maroon">A</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 ;<br/>

<a NAME="E4:20"/><b>then </b>
for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> holds <br/><font color="Olive">B</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:20"/>
<font color="Maroon">X0</font> is <a href="tex_3.html#V2" target="_self">everywhere_dense</a>
 <b>by </b><i/>;<br/>


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