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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:14"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Olive">b<sub>1</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14"/><i><font color="Green">E16</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>2</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </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="E1:14_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E2:14_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i><a class="txt" href="wellord2.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="ref" href="wellord1.html#T32">WELLORD1:32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><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="E1:14_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:14_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</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#K3">field</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T19">WELLORD1:19</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:14_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">c<sub>6</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord2.html#E3:14"><i><font color="Green">E16</font></i></a>, <a class="txt" href="wellord2.html#E1:14_1"><i><font color="Green">E17</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E5:14_1_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="wellord2.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="wellord2.html#E1:14_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="wellord2.html#E2:14_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord1.html#T35">WELLORD1:35</a></i>;<br/><a NAME="E6:14_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="wellord2.html#E2:14_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E7:14_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="wellord2.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="wellord2.html#E1:14_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="wellord2.html#E2:14_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="wellord1.html#T37">WELLORD1:37</a></i>;<br/><b>hence </b><a NAME="E8:14_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">c<sub>7</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord2.html#E4:14_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="wellord2.html#E5:14_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>;<br/></div><b>end;</b></div><a NAME="E4:14_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Olive">b<sub>1</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord2.html#E2:14_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="wellord2.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E5:14_1"/>
<font color="Maroon">c<sub>3</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="wellord2.html#E3:14"><i><font color="Green">E16</font></i></a>, <a class="txt" href="wellord2.html#E1:14_1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:14"/><b>then </b>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="ref" href="wellord1.html#T41">WELLORD1:41</a></i>;<br/>
<a NAME="E6:14"/><b>then </b>
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#K3">field</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b<sub>1</sub></font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Olive">b<sub>2</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord2.html#E3:14"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:14"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>1</sub></font>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Olive">b<sub>1</sub></font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord2.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="ref" href="wellord2.html#T13" target="_self">Th13</a></i>;<br/>


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