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<b>given </b><font color="Maroon" title="c1">N</font> being   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a><b> such that </b><a NAME="E2:67"/><i><font color="Green" title="E40">A1</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c1">N</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_1.html#D7" target="_self" title="CARD_1:def.7">CARD_1:def 7</a><br/>

<a NAME="E3:67"/>
<font color="Maroon" title="c1">N</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="card_1.html#E2:67"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="card_1.html#E45"><i><font color="Green" title="E19">Lm1</font></i></a>, <a class="ref" href="card_1.html#T32" target="_self" title="CARD_1:th.32">Th32</a>, <a class="ref" href="card_1.html#T83" target="_self" title="CARD_1:th.83">Th83</a></i>;<br/>
<a NAME="E4:67"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c1">N</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> )
 
<b>by </b><i><a class="txt" href="card_1.html#E97"><i><font color="Green" title="E39">Lm2</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">n</font> = <font color="Maroon" title="c1">N</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="txt" href="card_1.html#E2:67"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="card_1.html#E45"><i><font color="Green" title="E19">Lm1</font></i></a>, <a class="ref" href="card_1.html#T32" target="_self" title="CARD_1:th.32">Th32</a>, <a class="ref" href="card_1.html#T83" target="_self" title="CARD_1:th.83">Th83</a></i>;<br/>
<a NAME="E6:67"/><i><font color="Green" title="E42">A5</font></i>: 
 <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T76" target="_self" title="CARD_1:th.76">Th76</a></i>;<br/>
<a NAME="E7:67"/><i><font color="Green" title="E43">A6</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" target="_self" title="CARD_1:def.5">Def5</a></i>;<br/>
<a NAME="E8:67"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" target="_self" title="CARD_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E9:67"/>
contradiction
 <b>by </b><i><a class="txt" href="card_1.html#E2:67"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="card_1.html#E4:67"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="card_1.html#E45"><i><font color="Green" title="E19">Lm1</font></i></a>, <a class="ref" href="card_1.html#T83" target="_self" title="CARD_1:th.83">Th83</a>, <a class="txt" href="card_1.html#E6:67"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="card_1.html#E7:67"><i><font color="Green" title="E43">A6</font></i></a></i>;<br/>


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