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<b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:75"/><i><font color="Green">E32</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:75"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:75"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 ;<br/>

<a NAME="E4:75"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:75"/><b>then </b>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:75"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E7:75"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>,<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/>
<a NAME="E8:75"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<a NAME="E9:75"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:75"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<b>hence </b><a NAME="E11:75"/>
contradiction
 <b>by </b><i><a class="ref" href="finseq_4.html#T9" target="_self">Th9</a>, <a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/>


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