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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">R</font> be  <a href="msualg_6.html#V4" target="_self">stable</a> <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s</font>, <font color="Maroon">b</font> be   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s</font>;<br/>









<b>assume </b><a NAME="E1:72"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D4">REWRITE1:def 4</a><br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="rewrite1.html#M1">RedSequence</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span><b> such that </b><br/><a NAME="E3:72"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <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">b</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a></i>;<br/>
<b>let </b><font color="Maroon">h</font> be    <a href="msualg_6.html#M1" target="_self">Endomorphism</a> of <font color="Maroon">A</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>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> implies <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> );<br/>
<a NAME="E4:72"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:72"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</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">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<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>assume </b><b>that </b><br/><a NAME="E1:72_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> implies <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> )
 <b>and </b><br/><a NAME="E2:72_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:72_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:72_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1"><b>suppose </b></a><a NAME="E1:72_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:72_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="msualg_6.html#T4" target="_self">Th4</a>, <a class="ref" href="rewrite1.html#T27">REWRITE1:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_2"><b>suppose </b></a><a NAME="E1:72_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:72_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:72_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:72_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><a NAME="E5:72_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K6">~</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:72_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">ppi</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">pj</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msualg_6.html#NM1">Element</a> of <font color="Maroon">A</font>,<font color="Maroon">s</font> <b>by </b><i><a class="txt" href="msualg_6.html#E4:72_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:72_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">ppi</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">pj</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">pj</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">ppi</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E2:72_1_1_2"><i><font color="Green">E57</font></i></a>, </i>;<br/><a NAME="E9:72_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">ppi</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">pj</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> or <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">pj</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">ppi</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T30">REWRITE1:30</a></i>;<br/><a NAME="E10:72_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">ppi</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">pj</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i>, , <a class="txt" href="msualg_6.html#E2:72_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="rewrite1.html#T32">REWRITE1:32</a></i>;<br/><b>hence </b><a NAME="E11:72_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T31">REWRITE1:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:72"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">i</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">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="msualg_6.html#T5" target="_self">Th5</a>, );<br/></i>
<a NAME="E7:72"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>hence </b><a NAME="E8:72"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">R</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="msualg_6.html#T4" target="_self">Th4</a>, </i>;<br/>


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