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<div class="add">

<b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:38"/>
( <font color="Maroon" title="c2">X</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> implies  ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_4.html#D1" target="_self" title="CARD_4:def.1">CARD_4:def 1</a><br/>

<b>hence </b><a NAME="E2:38_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> )
 <b>by </b><i><a class="ref" href="card_1.html#T28" title="CARD_1:th.28">CARD_1:28</a>, <a class="ref" href="card_1.html#T83" title="CARD_1:th.83">CARD_1:83</a>, <a class="ref" href="card_1.html#T84" title="CARD_1:th.84">CARD_1:84</a></i>;<br/>


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

<b>assume </b><a NAME="E2:38"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> )
 ;<br/>

<b>hence </b><a NAME="E3:38"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T28" title="CARD_1:th.28">CARD_1:28</a>, <a class="ref" href="card_1.html#T83" title="CARD_1:th.83">CARD_1:83</a>, <a class="ref" href="card_1.html#T84" title="CARD_1:th.84">CARD_1:84</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_4.html#D1" target="_self" title="CARD_4:def.1">CARD_4:def 1</a><br/>


</div>
