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

<b>let </b><font color="Maroon" title="c1">O</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>;<br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">E</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">F2</font></span>)</span>,<font color="Maroon" title="c5">A</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b2">F2</font>,<font color="Maroon" title="c5">A</font></span>)</span>;<br/>
<a NAME="E1:96"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>;<br/>

<b>assume </b><a NAME="E1:96_1"/><i><font color="Green" title="E74">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>hence </b> <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F2</font></span>)</span>,<font color="Maroon" title="c5">A</font> = 
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">O</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">F2</font></span>)</span>,<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>
<br/>.= 
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F2</font>,<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F2</font>,<font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F2</font>,<font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E1:96_1"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="group_9.html#D3" target="_self" title="GROUP_9:def.3">Def3</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E2:96"/><i><font color="Green" title="E74">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:96_2"/><i><font color="Green" title="E75">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">F1</font>, <font color="Maroon" title="c8">F2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>;<br/><b>assume </b><a NAME="E1:96_2_1"/><i><font color="Green" title="E76">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><b>then consider </b><font color="Maroon" title="c9">F1k</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c10">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font><b> such that </b><br/><a NAME="E3:96_2_1"/><i><font color="Green" title="E77">A6</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">F1k</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">o</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></i>;<br/><a NAME="E4:96_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">o</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E3:96_2_1"><i><font color="Green" title="E77">A6</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:96_2_1"/><b>then </b><i><font color="Green" title="E78">A7</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E1:96_2_1"><i><font color="Green" title="E76">A5</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><b>set </b><font color="Maroon" title="c11">F2k</font> = <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">o</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">F2</font>;<br/><b>thus </b> <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">F2</font></span>)</span>,<font color="Maroon" title="c5">A</font> = 
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F1k</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">o</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">F2</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="txt" href="group_9.html#E3:96_2_1"><i><font color="Green" title="E77">A6</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c9">F1k</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">o</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">F2</font></span>)</span>,<font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E5:96_2_1"><i><font color="Green" title="E78">A7</font></i></a>, <a class="txt" href="group_9.html#E1:96_2"><i><font color="Green" title="E75">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c9">F1k</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">o</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E82"><i><font color="Green" title="E71">Lm28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c9">F1k</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">o</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>,<font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E3:96_2_1"><i><font color="Green" title="E77">A6</font></i></a>, <a class="txt" href="group_9.html#E5:96_2_1"><i><font color="Green" title="E78">A7</font></i></a>, <a class="txt" href="group_9.html#E1:96_2"><i><font color="Green" title="E75">A4</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:96_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:96"/><i><font color="Green" title="E75">A8</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="group_9.html#E1:96"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="group_9.html#E2:96"><i><font color="Green" title="E74">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E5:96"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F1</font>
 
;<br/>
<b>hence </b><a NAME="E6:96"/>
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>,<font color="Maroon" title="c5">A</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c5">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E3:96"><i><font color="Green" title="E75">A8</font></i></a></i>;<br/>


</div>
