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

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



<b>set </b><font color="Maroon">f</font> =  <a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:42"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">a</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">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 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">w</font>
;<br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E2:42_1"/>
<span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:42"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] holds <br/> ex <font color="Olive">w</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:42_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>given </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E2:42_2"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> 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">w</font>
;<br/>

<a NAME="E4:42_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:42_2"/>
<font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="real_3.html#E1:42_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_3.html#T1" target="_self">Th1</a>, </i>;<br/>

<b>thus </b><a NAME="E6:42_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">w</font>]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:42"/>
<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="txt" href="real_3.html#E1:42"><i><font color="Green">E31</font></i></a>, <a class="txt" href="real_3.html#E2:42"><i><font color="Green">E32</font></i></a>);</i><br/>


</div>
