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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">n</font>);<br/>
<a NAME="E1:24_1"/><i><font color="Green">E34</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i><a class="ref" href="scmfsa_9.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:24_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] )
 ;<br/>

<b>then 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="E3:24_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>)
 ;<br/>
<b>take </b>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)
;<br/>

<b>thus </b><a NAME="E4:24_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>(<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/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="hidden.html#R1">=</a> 0;<br/>
<a NAME="E3:24_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>from </b><i><a class="ref" href="nat_1.html#S7">NAT_1:sch 7</a>(, <a class="ref" href="scmfsa_9.html#T2" target="_self">Th2</a>);<br/></i>
<a NAME="E4:24_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<b>thus </b><a NAME="E5:24_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();</i><br/>


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