<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:10"/>
( not <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a> implies  ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Olive">Y</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/>
not <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:10_1"/><b>then </b>
not  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_1"/><b>then </b>
not  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i>, <a class="ref" href="card_1.html#T83">CARD_1:83</a></i>;<br/>
<a NAME="E4:10_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/>
<a NAME="E5:10_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:10_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>take </b><font color="Maroon">Y</font> = <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>;<br/>

<b>thus </b><a NAME="E8:10_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>

<a NAME="E9:10_1"/>
 <a href="card_1.html#K3">alef</a> 0,<font color="Maroon">Y</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E1:10_1_1"/>
( <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0 &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:10_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>


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

<b>given </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:10"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0 )
 ;<br/>

<a NAME="E3:10"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">Y</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E4:10"/><b>then </b>
not  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i>, <a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/>
<b>hence </b><a NAME="E5:10"/>
not <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/>


</div>
