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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">T</font> is <a href="topgen_1.html#V7">separable</a>
 ;<br/><a NAME="E2:13_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="topgen_1.html#K4">density</a> <font color="Maroon">T</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 <b>by </b><i><a class="ref" href="topgen_1.html#D13">TOPGEN_1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a> &amp;  <a href="topgen_1.html#K4">density</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="topgen_1.html#D12">TOPGEN_1:def 12</a></i>;<br/><a NAME="E5:13_1"/>
<font color="Maroon">A</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i>, , <a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/><b>hence </b><a NAME="E6:13_1"/>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> is <a href="tops_1.html#V1">dense</a> &amp; <font color="Olive">A</font> is <a href="card_4.html#V1">countable</a> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E2:13"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a> &amp; <font color="Maroon">A</font> is <a href="card_4.html#V1">countable</a> )
 ;<br/>

<a NAME="E3:13"/><i><font color="Green">E52</font></i>: 
 <a href="topgen_1.html#K4">density</a> <font color="Maroon">T</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="topgen_1.html#D12">TOPGEN_1:def 12</a></i>;<br/>
<a NAME="E4:13"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i>, <a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/>
<a NAME="E5:13"/><b>then </b>
 <a href="topgen_1.html#K4">density</a> <font color="Maroon">T</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">T</font> is <a href="topgen_1.html#V7">separable</a>
 <b>by </b><i><a class="ref" href="topgen_1.html#D13">TOPGEN_1:def 13</a></i>;<br/>


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