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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:58"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="card_1.html#K1" target="_self">Card</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> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:58"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="card_1.html#K1" target="_self">Card</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="card_1.html#NR2" target="_self">&lt;`</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>
 
<b>by </b><i/>;<br/>
<a NAME="E3:58"/>
for <font color="Olive">M</font> being  <a href="card_1.html#NM1">Cardinal</a>  st  <a href="card_1.html#K1" target="_self">Card</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="card_1.html#NR2" target="_self">&lt;`</a> <font color="Olive">M</font> holds <br/> <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="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Olive">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>

<b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E34</font></i>: 
 <a href="card_1.html#K1" target="_self">Card</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#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:58_1"/>
(  <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; <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:58_1"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">M</font> &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;  <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 )
 
<b>by </b><i>, , , <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<b>hence </b><a NAME="E4:58_1"/>
 <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="ordinal1.html#R1">c=</a> <font color="Maroon">M</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
 <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>
 <b>by </b><i>, </i>;<br/>


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