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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="E2:68"/><i><font color="Green" title="E44">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="E3:68"/><i><font color="Green" title="E45">A2</font></i>: 
<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#E2:68"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="card_1.html#E46"><i><font color="Green" title="E20">Lm2</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="E4:68"/><b>then </b><i><font color="Green" title="E46">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="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>, <a class="txt" href="card_1.html#E98"><i><font color="Green" title="E43">Lm3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:68"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c1">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="txt" href="card_1.html#E3:68"><i><font color="Green" title="E45">A2</font></i></a></i>;<br/>
<a NAME="E7:68"/>
(  <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"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <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> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">N</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T52" target="_self" title="CARD_1:th.52">Th52</a>, <a class="ref" href="card_1.html#D5" target="_self" title="CARD_1:def.5">Def5</a>, <a class="ref" href="card_1.html#T76" target="_self" title="CARD_1:th.76">Th76</a></i>;<br/>
<b>hence </b><a NAME="E8:68"/>
contradiction
 <b>by </b><i><a class="txt" href="card_1.html#E2:68"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="card_1.html#E4:68"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="card_1.html#E6:68"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="card_1.html#E46"><i><font color="Green" title="E20">Lm2</font></i></a>, <a class="ref" href="card_1.html#T83" target="_self" title="CARD_1:th.83">Th83</a></i>;<br/>


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