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

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span> ;<br/>
<a NAME="E1:51_1_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span> ,<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></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="E3:51_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:51_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span> 
 <b>and </b><br/><a NAME="E5:51_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span> 
 <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="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i><a class="txt" href="lattice5.html#E4:51_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<a NAME="E7:51_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K4">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:51_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K5">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K5">]</a></span> &amp; <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:51_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K4">:]</a></span>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:51_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="ordinal1.html#M1">T-Sequence</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K4">:]</a></span> <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E9:51_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> is   <a href="card_1.html#NM1">Cardinal</a>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:51_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:51_1_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E11:51_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">d</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div>
