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



<b>thus </b><a NAME="E1:16"/>
(  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font> implies  ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E33</font></i>: 
 <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">X</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:16_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">X</font>, <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">X</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">Y</font>, <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font> <a href="wellord2.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="E4:16_1"/><i><font color="Green">E42</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> <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>  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:16_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>take </b><font color="Maroon">h</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>;<br/>

<a NAME="E7:16_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/>
<b>hence </b><a NAME="E8:16_1"/>
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/>

<a NAME="E9:16_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, , , <a class="ref" href="relat_1.html#T45">RELAT_1:45</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>


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

<b>given </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E2:16"/><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> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E3:16"/>
<font color="Maroon">Y</font>, <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:16"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">X</font>, <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
; <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="E1:16_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <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#T46">FUNCT_1:46</a></i>;<br/>

<b>thus </b><a NAME="E2:16_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>

<b>thus </b><a NAME="E3:16_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:16"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E8:16"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="card_1.html#K1" target="_self">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:16"/>
( <font color="Maroon">X</font>, <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">X</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>, <a href="card_1.html#K1" target="_self">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E10:16"/><b>then </b>
(  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">X</font>, <a href="card_1.html#K1" target="_self">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i>, <a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
 <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">X</font>, <a href="card_1.html#K1" target="_self">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<b>hence </b><a NAME="E12:16"/>
 <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div>
