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

<b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D'</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D'</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">T</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">T'</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D'</font>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">S'</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D'</font>;<br/>



















<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>for <font color="Olive">S</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><br/> for <font color="Olive">S'</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D'</font> holds  <font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">S'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">T</font>,<font color="Maroon">T'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Olive">S</font>,<font color="Olive">S'</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">S'</font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D'</font>;<br/><a NAME="E1:20_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T113">FINSEQ_2:113</a></i>;<br/><a NAME="E2:20_1"/><b>then </b>
( <font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> &amp; <font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T'</font> &amp; <font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">S</font>,<font color="Maroon">S'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">E</font> &amp;  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#T87">FINSEQ_2:87</a>, <a class="ref" href="finseq_2.html#T115">FINSEQ_2:115</a></i>;<br/><b>hence </b><a NAME="E3:20_1"/>
<font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">T</font>,<font color="Maroon">T'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">S</font>,<font color="Maroon">S'</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:20"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">S</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><br/> for <font color="Olive">S'</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D'</font> holds  <font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">S'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">T</font>,<font color="Maroon">T'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Olive">S</font>,<font color="Olive">S'</font></span>)</span>
 ;<br/><b>let </b><font color="Maroon">S</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">S'</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D'</font>;<br/><b>consider </b><font color="Maroon">S1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/><b>consider </b><font color="Maroon">S1'</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D'</font>, <font color="Maroon">d'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D'</font><b> such that </b><br/><a NAME="E5:20_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d'</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/><a NAME="E6:20_2"/>
( <font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S1</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S1'</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D'</font> &amp; <font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S1'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a>, <a class="ref" href="finseq_2.html#T127">FINSEQ_2:127</a></i>;<br/><b>hence </b><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S'</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S1'</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">T</font>,<font color="Maroon">T'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">S1</font>,<font color="Maroon">S1'</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">T</font>,<font color="Maroon">T'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">S1</font>,<font color="Maroon">S1'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">T</font>,<font color="Maroon">T'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">S</font>,<font color="Maroon">S'</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E4:20"/><b>then </b><i><font color="Green">E51</font></i>: 
for <font color="Olive">j</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">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:20"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E6:20"/>
<font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">T'</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">S'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">T</font>,<font color="Maroon">T'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseqop.html#K1" target="_self">.:</a> <font color="Maroon">S</font>,<font color="Maroon">S'</font></span>)</span>
 ;<br/>


</div>
