<?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="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a>  &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a> )
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="classes1.html#V1">subset-closed</a>
 
<b>by </b><i><a class="ref" href="classes1.html#D2">CLASSES1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<b>assume </b><a NAME="E3:26"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E4:26"/><b>then </b><i><font color="Green">E25</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>and </b><br/><a NAME="E7:26"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="txt" href="classes2.html#E4:26"><i><font color="Green">E25</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:26"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="classes2.html#E6:26"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E10:26"/><i><font color="Green">E29</font></i>: 
(  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E1:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="classes2.html#E9:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="classes2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E11:26"/><i><font color="Green">E30</font></i>: 
 <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E9:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1"/>
 <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E10:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E3:26_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:26_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:26_1_1"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E9:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="classes1.html#T76">CLASSES1:76</a></i>;<br/>
<a NAME="E3:26_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E10:26"><i><font color="Green">E29</font></i></a>, <a class="txt" href="classes2.html#E1:26_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<a NAME="E4:26_1_1"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E7:26"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E5:26_1_1"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E10:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E6:26_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="classes1.html#T42">CLASSES1:42</a>, <a class="ref" href="classes1.html#D8">CLASSES1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:26_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T47">CLASSES1:47</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E9:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="classes2.html#E10:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="classes1.html#T73">CLASSES1:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:26"/><i><font color="Green">E31</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E2:26"><i><font color="Green">E24</font></i></a>, <a class="txt" href="classes2.html#E9:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="classes2.html#T1" target="_self">Th1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<b>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="E14:26"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E15:26"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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>thus </b><a NAME="E1:26_2"/>
(  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="classes2.html#E14:26"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:26_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:26_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E14:26"><i><font color="Green">E32</font></i></a>, <a class="txt" href="classes2.html#E3:26_3"><i><font color="Green">E34</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/><a NAME="E6:26_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E3:26_3"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E7:26_3"/>
<font color="Maroon">c<sub>6</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="classes2.html#E6:26_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E9:26_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="classes2.html#E3:26_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E10:26_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T31">ORDINAL1:31</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E19:26"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &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> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E20:26"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:26_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="classes2.html#E14:26"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E4:26_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E19:26"><i><font color="Green">E34</font></i></a>, <a class="txt" href="classes2.html#E3:26_4"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:26_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E19:26"><i><font color="Green">E34</font></i></a>, <a class="txt" href="classes2.html#E3:26_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:26"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E19:26"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_1.html#T28">CARD_1:28</a></i>;<br/>
<a NAME="E22:26"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E12:26"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a>, <a class="ref" href="ordinal2.html#T9">ORDINAL2:9</a></i>;<br/>
<a NAME="E23:26"/><b>then </b>
( not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E15:26"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E24:26"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E1:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="classes1.html#D2">CLASSES1:def 2</a></i>;<br/>
<a NAME="E25:26"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E1:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="classes2.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E26:26"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5"><b>proof </b></a><div class="add">

 <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
;<br/>
<b>hence </b><a NAME="E2:26_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E27:26"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#T54">CARD_3:54</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="card_3.html#NM1">Cardinal-Function</a><b> such that </b><br/><a NAME="E29:26"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &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> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="card_3.html#S1">CARD_3:sch 1</a>();<br/></i>
<a NAME="E30:26"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a>, <a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:26_6"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:26_6"/><b>then </b>
( <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="classes2.html#E29:26"><i><font color="Green">E37</font></i></a>, <a class="txt" href="classes2.html#E30:26"><i><font color="Green">E38</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E3:26_6"/>
<span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="card_1.html#T30">CARD_1:30</a></i>;<br/></div><b>end;</b></div>
<a NAME="E32:26"/><b>then </b>
 <a href="card_3.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="card_3.html#K1">Card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_3.html#K7">Product</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E29:26"><i><font color="Green">E37</font></i></a>, <a class="txt" href="classes2.html#E30:26"><i><font color="Green">E38</font></i></a>, <a class="ref" href="card_3.html#T56">CARD_3:56</a></i>;<br/>
<a NAME="E33:26"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_3.html#K7">Product</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E27:26"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E34:26"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:26_7"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:26_7"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:26_7"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</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="E1:26_7_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_7_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:26_7_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E15:26"><i><font color="Green">E33</font></i></a>, <a class="txt" href="classes2.html#E29:26"><i><font color="Green">E37</font></i></a>, <a class="txt" href="classes2.html#E3:26_7"><i><font color="Green">E40</font></i></a>, <a class="txt" href="classes2.html#E3:26_7_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E5:26_7_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E15:26"><i><font color="Green">E33</font></i></a>, <a class="txt" href="classes2.html#E29:26"><i><font color="Green">E37</font></i></a>, <a class="txt" href="classes2.html#E3:26_7"><i><font color="Green">E40</font></i></a>, <a class="txt" href="classes2.html#E3:26_7_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E6:26_7_1"/><b>then </b>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E1:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="classes1.html#D2">CLASSES1:def 2</a></i>;<br/>
<a NAME="E7:26_7_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E1:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="classes2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E8:26_7_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E1:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="classes2.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E9:26_7_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E4:26_7_1"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:26_7"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes2.html#E29:26"><i><font color="Green">E37</font></i></a>, <a class="txt" href="classes2.html#E3:26_7"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:26"/><b>then </b>
(  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K7">Product</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E25:26"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_3.html#D8">CARD_3:def 8</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:26"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="card_3.html#K7">Product</a> <font color="Maroon">c<sub>8</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E37:26"/>
 <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:26_8"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_8_1"/><i><font color="Green">E42</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> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="classes1.html#K6">the_rank_of</a> <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:26_8_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</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="E1:26_8_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:26_8_1_1"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E1:26_8_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E3:26_8_1_1"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<b>hence </b><a NAME="E4:26_8_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T74">CLASSES1:74</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:26_8_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="classes2.html#E9:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="classes2.html#E1:26_8"><i><font color="Green">E41</font></i></a>, <a class="ref" href="classes1.html#T73">CLASSES1:73</a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/><a NAME="E4:26_8_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="classes2.html#E1:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="classes2.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E5:26_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="classes2.html#E2:26"><i><font color="Green">E24</font></i></a>, <a class="txt" href="classes2.html#E9:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:26_8"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>
<a NAME="E6:26_8"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E9:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="classes2.html#E5:26_8"><i><font color="Green">E42</font></i></a>, <a class="ref" href="classes1.html#T76">CLASSES1:76</a></i>;<br/>
<a NAME="E7:26_8"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E11:26"><i><font color="Green">E30</font></i></a>, <a class="txt" href="classes2.html#E14:26"><i><font color="Green">E32</font></i></a>, <a class="txt" href="classes2.html#E5:26_8"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E8:26_8"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E5:26_8"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E9:26_8"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E38:26"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="classes2.html#E1:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="classes2.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E39:26"/>
contradiction
 <b>by </b><i><a class="txt" href="classes2.html#E33:26"><i><font color="Green">E39</font></i></a>, <a class="txt" href="classes2.html#E36:26"><i><font color="Green">E40</font></i></a>, <a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/>


</div>
