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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="classes1.html#K4">Rank</a> <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>;<br/>
<a NAME="E1:57"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="classes1.html#T33">CLASSES1:33</a></i>;<br/>
<a NAME="E2:57"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<b>by </b><i><a class="txt" href="card_3.html#E65"><i><font color="Green">Lemma25</font></i></a></i>;<br/>
<a NAME="E3:57"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="card_3.html#E1:57"><i><font color="Green">E26</font></i></a>, <a class="txt" href="card_3.html#E2:57"><i><font color="Green">E27</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:57"/>
 <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


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