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<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  implies <font color="Maroon" title="c1">x</font> is <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green" title="E34">A2</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/> ex <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">C</font>,<font color="Olive" title="b1">B</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; not <font color="Olive" title="b1">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">C</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_1.html#D1" target="_self" title="CARD_1:def.1">CARD_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c2">A</font> = <font color="Maroon" title="c1">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">A</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; not <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="#E2:49"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<a NAME="E6:49"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E5:49"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E7:49"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c3">B</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>
<a NAME="E9:49"/>
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">m</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="#E5:49"><i><font color="Green" title="E35">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:49"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:49"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="card_1.html#E74"><i><font color="Green" title="E31">Th64</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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