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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>thus </b><a NAME="E1:16"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font> 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/>( <font color="Olive" title="b1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <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> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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" title="E10">A1</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>

<a NAME="E2:16_1"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c1">X</font>, <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c2">Y</font>, <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#D5" target="_self" title="CARD_1:def.5">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:16_1"/><i><font color="Green" title="E12">A3</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:16_1"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c4">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="card_1.html#E2:16_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>;<br/>

<a NAME="E7:16_1"/>
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="card_1.html#E6:16_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="funct_1.html#T62" title="FUNCT_1:th.62">FUNCT_1:62</a></i>;<br/>
<b>hence </b><a NAME="E8:16_1"/>
<font color="Maroon" title="c5">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_1.html#E4:16_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>;<br/>

<a NAME="E9:16_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="card_1.html#E6:16_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="card_1.html#E1:16_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="card_1.html#E4:16_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="card_1.html#E6:16_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E3:16"/><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<a NAME="E4:16"/>
<font color="Maroon" title="c2">Y</font>, <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5" target="_self" title="CARD_1:def.5">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:16"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c4">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E7:16"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c1">X</font>, <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">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" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E1:16_2"/>
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_1.html#E3:16"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="card_1.html#E6:16"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>;<br/>

<b>thus </b><a NAME="E2:16_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="card_1.html#E3:16"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="card_1.html#E6:16"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>

<b>thus </b><a NAME="E3:16_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:16"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="card_1.html#E6:16"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T23" target="_self" title="CARD_1:th.23">Th23</a></i>;<br/>
<a NAME="E10:16"/>
( <font color="Maroon" title="c1">X</font>, <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span>, <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#D5" target="_self" title="CARD_1:def.5">Def5</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c1">X</font>, <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="card_1.html#E7:16"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="wellord2.html#T22" title="WELLORD2:th.22">WELLORD2:22</a></i>;<br/>
<a NAME="E12:16"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font>, <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#T22" title="WELLORD2:th.22">WELLORD2:22</a></i>;<br/>
<b>hence </b><a NAME="E13:16"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="card_1.html#E9:16"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="card_1.html#T8" target="_self" title="CARD_1:th.8">Th8</a></i>;<br/>


</div>
