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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>];<br/>
<a NAME="E1:9_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E2:9_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/>for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:9_1"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S5" target="_self">RECDEF_1:sch 5</a>(, );<br/></i>
<a NAME="E5:9_1"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>take </b>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E2:9_1_1"/>
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i>, </i>;<br/>

<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><a NAME="E3:9_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/>

<b>hence </b><a NAME="E4:9_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:9_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> 0 implies  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>take </b>
<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 0
;<br/>

<b>take </b>
<font color="Maroon">F<sub>1</sub></font>()
;<br/>

<b>thus </b><a NAME="E2:9_1_2"/>
( <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E3:9_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:9_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">p</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>by </b><i/>;<br/>


</div>
