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<span class="kw">let </span><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c3">K</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"> ( ( not <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font> &amp; not <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">N</font> ) or <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or (  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">N</font> &amp;  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">K</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">K</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:167"/><span class="lab"><font color="Green" title="E80">A1</font></span>: 
( <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font> or <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">N</font> )
 <span class="kw">and </span><br/><a NAME="E2:167"/><span class="lab"><font color="Green" title="E81">A2</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">N</font> &amp;  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">K</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">K</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:167"/>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="txt" href="#E2:167"><span class="lab"><font color="Green" title="E81">A2</font></span></a>, <a class="ref" href="card_3.html#T138" target="_self" title="CARD_3:th.138">Th138</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">K</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">K</font></span><br/>

<a NAME="E4:167"/>
( <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:167"><span class="lab"><font color="Green" title="E81">A2</font></span></a>, <a class="ref" href="card_2.html#T39" title="CARD_2:th.39">CARD_2:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:167"/>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">K</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:167"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="ref" href="card_3.html#T138" target="_self" title="CARD_3:th.138">Th138</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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