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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>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topgen_2.html#K1" target="_self">Chi</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E1:9"/><i><font color="Green">E9</font></i>: 
 <a href="topgen_2.html#K2" target="_self">Chi</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topgen_2.html#K1" target="_self">Chi</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="topgen_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>thus </b><a NAME="E2:9"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="frechet.html#V1">first-countable</a> implies  <a href="topgen_2.html#K2" target="_self">Chi</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> is <a href="card_4.html#V1">countable</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D1">FRECHET:def 1</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topgen_2.html#K1" target="_self">Chi</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:9_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="topgen_2.html#K1" target="_self">Chi</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E5:9_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_2.html#E1:9_1"><i><font color="Green">E10</font></i></a></i>;<br/><a NAME="E6:9_1_1"/>
(  <a href="topgen_2.html#K1" target="_self">Chi</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></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="txt" href="topgen_2.html#E5:9_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="topgen_2.html#D1" target="_self">Def1</a>, <a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/><b>hence </b><a NAME="E7:9_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K3">alef</a> 0
 <b>by </b><i><a class="txt" href="topgen_2.html#E3:9_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1"/>
 <a href="topgen_2.html#K2" target="_self">Chi</a> <font color="Maroon">c<sub>1</sub></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="txt" href="topgen_2.html#E1:9"><i><font color="Green">E9</font></i></a>, <a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:9"/><i><font color="Green">E10</font></i>: 
 <a href="topgen_2.html#K2" target="_self">Chi</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D1">FRECHET:def 1</a><br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:9"/><i><font color="Green">E11</font></i>: 
 <a href="topgen_2.html#K1" target="_self">Chi</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="topgen_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E6:9"/>
 <a href="topgen_2.html#K1" target="_self">Chi</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="topgen_2.html#K2" target="_self">Chi</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="topgen_2.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></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="txt" href="topgen_2.html#E3:9"><i><font color="Green">E10</font></i></a>, <a class="txt" href="topgen_2.html#E5:9"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_4.html#D1">CARD_4:def 1</a><br/>


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