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

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">O1</font>, <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Olive">O2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">O1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Olive">O2</font> );<br/>
<a NAME="E1:15"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:15"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">x2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">x2</font>
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>thus </b><a NAME="E4:15_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">x2</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:15"/><i><font color="Green">E40</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E5:15"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>take </b>
<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="E1:15_2"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:15_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">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="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:15_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">x2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">y1</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">y2</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">O1</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">O2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:15_2_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">O2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">O1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">O2</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">O3</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">O4</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:15_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">O3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">O4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">O3</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">O4</font> )
 <b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E8:15_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E9:15_2_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 
<b>by </b><i>, , , <a class="ref" href="ordinal3.html#T56">ORDINAL3:56</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:15_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E3:15_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:15_2_2"/>
<font color="Maroon">y</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="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_2_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">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="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:15_2_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">x2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:15_2_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:15_2_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">x2</font> &amp; 1 <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x1</font> <a href="ordinal2.html#K10">+^</a> 1 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x1</font> )
 
<b>by </b><i>, , <a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<a NAME="E8:15_2_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default">1 <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default">1 <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">x1</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a>, <a class="ref" href="ordinal3.html#T54">ORDINAL3:54</a></i>;<br/>
<a NAME="E9:15_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default">1 <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T58">ORDINAL2:58</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">y</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="E4:15_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E6:15_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font></span>)</span> &amp; <font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal3.html#T78">ORDINAL3:78</a>, <a class="ref" href="ordinal3.html#T80">ORDINAL3:80</a></i>;<br/>
<a NAME="E7:15_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">O1</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">O2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:15_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">O1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">O2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">O1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">O2</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E10:15_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="wellord2.html#R2">are_equipotent</a> 
 ;<br/>

<b>thus </b><a NAME="E7:15"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
