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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E10</font></i>: 
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</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>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E3:13"/><i><font color="Green">E12</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>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:13"><i><font color="Green">E11</font></i></a></i>;<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="E5:13"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>and </b><br/><a NAME="E6:13"/><i><font color="Green">E14</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>1</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="ordinal4.html#E3:13"><i><font color="Green">E12</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<a NAME="E8:13"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</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="ordinal4.html#E5:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<a NAME="E9:13"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="ordinal4.html#E5:13"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E10:13"/><b>then </b>
( <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>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E5:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="ordinal4.html#E8:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E11:13"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="ordinal4.html#E6:13"><i><font color="Green">E14</font></i></a></i>;<br/>


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