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<b>assume </b><a NAME="E1:35"/><i><font color="Green">E23</font></i>: 
 <a href="card_5.html#K1" target="_self">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K3">alef</a> 0
 ;<br/>

<a NAME="E2:35"/>
 <a href="card_5.html#K1" target="_self">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:35"/><b>then </b>
 <a href="card_5.html#K1" target="_self">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i><a class="txt" href="card_5.html#E8:35"><i><font color="Green">E24</font></i></a>, <a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> =  <a href="card_5.html#K1" target="_self">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_4.html#T9">CARD_4:9</a></i>;<br/>
<b>set </b><font color="Maroon">n</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font>;<br/>
<a NAME="E5:35"/>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_5.html#K1" target="_self">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">B</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#K4">card</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" target="_self">cf</a> <span class="p2">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span></span>)</span> <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#K4">card</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E6:35"/><b>then </b>
 <a href="card_1.html#K3">alef</a> 0 <a href="zfrefle1.html#R4">is_cofinal_with</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">xi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E8:35"/><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K3">alef</a> 0 &amp; <font color="Maroon">xi</font> is <a href="ordinal2.html#V2">increasing</a> &amp;  <a href="card_1.html#K3">alef</a> 0 <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font> )
 <b>by </b><i><a class="ref" href="zfrefle1.html#D5">ZFREFLE1:def 5</a></i>;<br/>
<a NAME="E9:35"/><i><font color="Green">E25</font></i>: 
(  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font></span>)</span> &amp;  <a href="ordinal2.html#K3">sup</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="card_1.html#K3">alef</a> 0 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T83">CARD_1:83</a>, <a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rxi</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="card_5.html#E9:35"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<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><font color="Maroon">a<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E11:35"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">rxi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_5.html#E9:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="card_5.html#E11:35"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E12:35"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font>, <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>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E13:35"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</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>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:35"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rxi</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rxi</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> holds <br/> not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Maroon">x</font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S2">CARD_3:sch 2</a>(, , );<br/></i>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E9:35"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font>
 ;<br/><a NAME="E2:35_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">x</font> or  not <font color="Maroon">x</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:35_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:35"/><b>then </b>
(  <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 
<b>by </b><i><a class="txt" href="card_5.html#E9:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="card_5.html#E11:35"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a>, <a class="ref" href="ordinal2.html#T28">ORDINAL2:28</a></i>;<br/>
<b>hence </b><a NAME="E19:35"/>
contradiction
 <b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


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