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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="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</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"> ( <font color="Maroon" title="c1">T</font> is <a href="topgen_1.html#V7" title="TOPGEN_1:attr.7">separable</a> iff  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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>  )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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>  implies <font color="Maroon" title="c1">T</font> is <a href="topgen_1.html#V7" title="TOPGEN_1:attr.7">separable</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon" title="c1">T</font> is <a href="topgen_1.html#V7" title="TOPGEN_1:attr.7">separable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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> </span><br/><a NAME="E2:18_1"/><b>then </b><i><font color="Green" title="E7">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="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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="E4:18_1"/><i><font color="Green" title="E8">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/><b>thus </b><a NAME="E5:18_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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#E2:18_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="topgen_4.html#E4:18_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="tops_3.html#T19" title="TOPS_3:th.19">TOPS_3:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:18"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T</font> is <a href="topgen_1.html#V7" title="TOPGEN_1:attr.7">separable</a></span><br/>

<a NAME="E3:18"/><b>then </b>
<font color="Maroon" title="c1">T</font> is <a href="orders_4.html#V1" title="ORDERS_4:attr.1">countable</a>
 
<b>by </b><i><a class="ref" href="topgen_1.html#T3" title="TOPGEN_1:th.3">TOPGEN_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
<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#T49" title="TOPGEN_1:th.49">TOPGEN_1:49</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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