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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="card_4.html#V1">countable</a>  <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1"/>
( <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a> or  not <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>suppose </b></a><a NAME="E1:15_1_1"/>
<font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:15_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="msualg_9.html#T3">MSUALG_9:3</a></i>;<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><b>thus </b><a NAME="E4:15_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2"><b>suppose </b></a><a NAME="E1:15_1_2"/>
not <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><div class="add"><a NAME="E2:15_1_2"/><b>then </b>
 <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><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="E4:15_1_2"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:15_1_2"/><i><font color="Green">E26</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><b>take </b>
<font color="Maroon">f</font>
;<br/><b>thus </b><a NAME="E7:15_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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