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<span class="kw">let </span><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">N</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> iff <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:13"/>
( <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> implies <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  implies <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:13"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E4:13"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c3">A</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  holds <br/><font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D1" target="_self" title="CARD_1:def.1">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E6:13"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E7:13"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c4">B</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  holds <br/><font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D1" target="_self" title="CARD_1:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:13"/>
<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font></span><br/>

<a NAME="E9:13"/><span class="kw">then </span>
( <font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E6:13"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E7:13"><span class="lab"><font color="Green" title="E7">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:13"/>
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E6:13"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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