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<b>given </b><font color="Maroon">N</font> being   <a href="card_1.html#NM1">Cardinal</a><b> such that </b><a NAME="E1:67"/><i><font color="Green">E33</font></i>: 
 <a href="card_1.html#K1" target="_self">Card</a> <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2" target="_self">nextcard</a> <font color="Maroon">N</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_1.html#D7" target="_self">CARD_1:def 7</a><br/>

<a NAME="E2:67"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="txt" href="card_1.html#E5:67"><i><font color="Green">E43</font></i></a>, , , <a class="txt" href="card_1.html#E3:67"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E3:67"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">N</font> is <a href="ordinal1.html#V7">natural</a> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:67"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="card_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E6:67"/>
(  <a href="card_1.html#K2" target="_self">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1" target="_self">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="card_1.html#K1" target="_self">Card</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E7:67"/>
contradiction
 <b>by </b><i><a class="txt" href="card_1.html#E5:67"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_1.html#D5" target="_self">Def5</a>, <a class="ref" href="card_1.html#T21" target="_self">Th21</a>, , <a class="txt" href="card_1.html#E3:67"><i><font color="Green">E42</font></i></a></i>;<br/>


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