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<b>let </b><font color="Maroon" title="c1">T</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>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c1">T</font> is <a href="topgen_1.html#V7" title="TOPGEN_1:attr.7">separable</a>
 ;<br/><a NAME="E2:13_1"/><b>then </b><i><font color="Green" title="E5">A1</font></i>: 
 <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">T</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="topgen_1.html#D13" title="TOPGEN_1:def.13">TOPGEN_1:def 13</a></i>;<br/><b>consider </b><font color="Maroon" title="c2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp;  <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="topgen_1.html#D12" title="TOPGEN_1:def.12">TOPGEN_1:def 12</a></i>;<br/><a NAME="E5:13_1"/>
<font color="Maroon" title="c2">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:13_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="topgen_4.html#E4:13_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="card_4.html#D1" title="CARD_4:def.1">CARD_4:def 1</a></i>;<br/><b>hence </b><a NAME="E6:13_1"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Olive" title="b1">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:13_1"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E3:13"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> )
 ;<br/>

<a NAME="E4:13"/><i><font color="Green" title="E6">A4</font></i>: 
 <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">T</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E3:13"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="topgen_1.html#D12" title="TOPGEN_1:def.12">TOPGEN_1:def 12</a></i>;<br/>
<a NAME="E5:13"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">A</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="topgen_4.html#E3:13"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="card_4.html#D1" title="CARD_4:def.1">CARD_4:def 1</a></i>;<br/>
<a NAME="E6:13"/><b>then </b>
 <a href="topgen_1.html#K4" title="TOPGEN_1:func.4">density</a> <font color="Maroon" title="c1">T</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="topgen_4.html#E4:13"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:13"/>
<font color="Maroon" title="c1">T</font> is <a href="topgen_1.html#V7" title="TOPGEN_1:attr.7">separable</a>
 <b>by </b><i><a class="ref" href="topgen_1.html#D13" title="TOPGEN_1:def.13">TOPGEN_1:def 13</a></i>;<br/>


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