<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1];<br/>
<b>consider </b><font color="Maroon">t</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>]
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E7:1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">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/>;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E4:1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E5:1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k1</font>
 <b>and </b><br/><a NAME="E8:1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>]
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>take </b><font color="Maroon">l</font> = <font color="Maroon">l</font>;<br/><b>thus </b><a NAME="E12:1_1_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><a NAME="E13:1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">l</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E9:1_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>(, );<br/></i>
<b>hence </b><a NAME="E10:1_1"/>
<font color="Maroon">P<sub>1</sub></font>[1]
 ;<br/>


</div>
