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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:90"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel23.html#V5">second-countable</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:90"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel23.html#K2">weight</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="topgen_2.html#T20">TOPGEN_2:20</a></i>;<br/>
<a NAME="E5:90"/><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E6:90"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="txt" href="topgen_4.html#E4:90"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topgen_4.html#E1:90"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel23.html#D6">WAYBEL23:def 6</a></i>;<br/>
<a NAME="E7:90"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E8:90"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="ref" href="waybel12.html#T2">WAYBEL12:2</a>, <a class="ref" href="card_4.html#T44">CARD_4:44</a></i>;<br/>
<a NAME="E9:90"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E5:90"><i><font color="Green">E50</font></i></a>, <a class="ref" href="yellow_8.html#T5">YELLOW_8:5</a></i>;<br/>
<b>hence </b><a NAME="E10:90"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="card_4.html#V1">countable</a>
 ;<br/>


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