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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></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">b<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E1:15"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
;<br/>
<a NAME="E2:15"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></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<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E4:15_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:15"/><i><font color="Green">E14</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="card_2.html#E1:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="card_2.html#E2:15"><i><font color="Green">E13</font></i></a>);<br/></i>
<a NAME="E5:15"/><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:15_2_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:15_2_1"/><b>then </b><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>4</sub></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<sub>4</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>5</sub></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<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_2.html#E4:15"><i><font color="Green">E14</font></i></a>, <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">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></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">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:15_2_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="card_2.html#E4:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="card_2.html#E1:15_2_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:15_2_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="card_2.html#E4:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="card_2.html#E1:15_2_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>assume </b><a NAME="E8:15_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E9:15_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="card_2.html#E2:15_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="card_2.html#E5:15_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="card_2.html#E7:15_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal3.html#T56">ORDINAL3:56</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="card_2.html#E2:15_2_1"><i><font color="Green">E17</font></i></a></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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="card_2.html#E4:15"><i><font color="Green">E14</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:15_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></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">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_2_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></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">E17</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_2.html#E4:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="card_2.html#E3:15_2_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:15_2_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="card_2.html#E4:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="card_2.html#E3:15_2_2"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E7:15_2_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font> &amp; 1 <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K10">+^</a> 1 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="card_2.html#E3:15_2_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="card_2.html#E6:15_2_2"><i><font color="Green">E18</font></i></a>, <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">E19</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="card_2.html#E4:15_2_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="card_2.html#E6:15_2_2"><i><font color="Green">E18</font></i></a>, <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">c<sub>6</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_2.html#E8:15_2_2"><i><font color="Green">E19</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></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">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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">E17</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_2.html#E4:15_2"><i><font color="Green">E16</font></i></a>, <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">E18</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></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<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></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<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:15_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>6</sub></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<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>7</sub></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<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></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<sub>5</sub></font> <a href="ordinal3.html#K6">div^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal3.html#K7">mod^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="card_2.html#E4:15"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:15_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_2.html#E4:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="card_2.html#E6:15_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="card_2.html#E7:15_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="card_2.html#E9:15_2"><i><font color="Green">E19</font></i></a>, <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">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="card_2.html#E5:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
