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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E2:39"/><i><font color="Green">E28</font></i>: 
(  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="zfrefle1.html#R4">is_cofinal_with</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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="card_5.html#E1:39"><i><font color="Green">E27</font></i></a>, <a class="ref" href="card_5.html#T15" target="_self">Th15</a>, <a class="ref" href="card_5.html#T16" target="_self">Th16</a>, <a class="ref" href="wellord2.html#T17">WELLORD2:17</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="card_1.html#NM1">Cardinal</a><b> such that </b><br/><a NAME="E4:39"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp;  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="zfrefle1.html#R4">is_cofinal_with</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="zfrefle1.html#R4">is_cofinal_with</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="card_5.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E5:39"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="zfrefle1.html#R4">is_cofinal_with</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E1:39"><i><font color="Green">E27</font></i></a>, <a class="txt" href="card_5.html#E2:39"><i><font color="Green">E28</font></i></a>, <a class="txt" href="card_5.html#E4:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a>, <a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a>, <a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a>, <a class="ref" href="zfrefle1.html#T21">ZFREFLE1:21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/>
 <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:39_1"/><b>then </b>
 <a href="card_5.html#K1" target="_self">cf</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="card_5.html#E5:39"><i><font color="Green">E30</font></i></a>, <a class="ref" href="card_5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E3:39_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_5.html#E1:39"><i><font color="Green">E27</font></i></a>, <a class="txt" href="card_5.html#E2:39"><i><font color="Green">E28</font></i></a>, <a class="txt" href="card_5.html#E4:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:39"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_5.html#E5:39"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:39"/><i><font color="Green">E31</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>

<a NAME="E9:39"/>
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>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<b>by </b><i><a class="txt" href="card_5.html#E1:39"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T35">ORDINAL1:35</a></i>;<br/>
<a NAME="E11:39"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_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>assume </b><a NAME="E1:39_2"/>
<font color="Maroon">c<sub>6</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>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_2"/><i><font color="Green">E32</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>1</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>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="txt" href="card_5.html#E1:39"><i><font color="Green">E27</font></i></a>, <a class="txt" href="card_5.html#E3:39_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:39_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_5.html#E3:39_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E6:39_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:39_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_5.html#E3:39_2"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:39"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</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_5.html#E8:39"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div>
