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<div class="add">

<b>let </b><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">T</font> be   <a href="rewrite2.html#NM1" title="REWRITE2:NM.1">semi-Thue-system</a> of <font color="Maroon" title="c1">E</font>;<br/><b>let </b><font color="Maroon" title="c4">w</font>, <font color="Maroon" title="c5">s</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green" title="E53">A1</font></i>: 
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">T</font>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green" title="E54">A2</font></i>: 
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">s</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite2.html#D9" target="_self" title="REWRITE2:def.9">REWRITE2:def 9</a><br/>

<b>consider </b><font color="Maroon" title="c6">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font></span>)</span><b> such that </b><br/><a NAME="E4:82"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:82"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">p</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font></span>)</span><br/> for <font color="Olive" title="b2">t</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> holds <br/> ex <font color="Olive" title="b3">q</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> );<br/>
<a NAME="E5:82"/><i><font color="Green" title="E56">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/>
<div><i><font color="Green" title="E57">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:82_1"/><i><font color="Green" title="E58">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 ;<br/><b>thus </b><a NAME="E2:82_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">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="82_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c9">t</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> ;<br/>



<b>assume </b><a NAME="E1:82_1_1"/><i><font color="Green" title="E59">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_1_1"/>
( ( not <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> ) or  not <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> or ( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_1"/>
( not <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> )
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="rewrite2.html#T1" target="_self" title="REWRITE2:th.1">Th1</a></i>;<br/><a NAME="E3:82_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:82_1_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c8">p</font> as    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="rewrite1.html#T7" title="REWRITE1:th.7">REWRITE1:7</a></i>;<br/><b>take </b>
<font color="Maroon" title="c10">p</font>
;<br/><b>thus </b><a NAME="E5:82_1_1_1_1"/>
( <font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c10">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:82_1_1"><i><font color="Green" title="E59">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_2"/>
not <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font>
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:82_1_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E3:82_1_1_1_2"/>
 ex <font color="Olive" title="b1">q</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_3"><b>suppose </b></a><a NAME="E1:82_1_1_1_3"/><i><font color="Green" title="E60">A8</font></i>: 
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> )
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_3"/><b>then </b><i><font color="Green" title="E61">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>;<br/><b>reconsider </b><font color="Maroon" title="c11">u</font> = <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> as    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  <b>by </b><i><a class="txt" href="rewrite2.html#E2:82_1_1_1_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:82_1_1_1_3"/><i><font color="Green" title="E62">A10</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:82_1_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="txt" href="rewrite2.html#E2:82_1_1_1_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">p1</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font></span>)</span><b> such that </b><br/><a NAME="E6:82_1_1_1_3"/><i><font color="Green" title="E63">A11</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c12">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c12">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c12">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:82_1_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="txt" href="rewrite2.html#E1:82_1_1_1_3"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="rewrite2.html#T4" target="_self" title="REWRITE2:th.4">Th4</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">q</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E8:82_1_1_1_3"/><i><font color="Green" title="E64">A12</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c13">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c13">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:82_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="txt" href="rewrite2.html#E6:82_1_1_1_3"><i><font color="Green" title="E63">A11</font></i></a></i>;<br/><a NAME="E9:82_1_1_1_3"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c11">u</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E8:82_1_1_1_3"><i><font color="Green" title="E64">A12</font></i></a>, <a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></i>;<br/><a NAME="E10:82_1_1_1_3"/><b>then </b><i><font color="Green" title="E65">A13</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D7" target="_self" title="REWRITE2:def.7">Def7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_1_1_3_1"/>
( <font color="Maroon" title="c11">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c11">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c3">T</font> )
 </span><b>by </b><i><a class="txt" href="rewrite2.html#E4:82_1_1_1_3"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="rewrite2.html#T21" target="_self" title="REWRITE2:th.21">Th21</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_3_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#T33" target="_self" title="REWRITE2:th.33">Th33</a></i>;<br/><a NAME="E3:82_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E10:82_1_1_1_3"><i><font color="Green" title="E65">A13</font></i></a>, <a class="ref" href="rewrite2.html#T35" target="_self" title="REWRITE2:th.35">Th35</a></i>;<br/><a NAME="E4:82_1_1_1_3_1_1"/><b>then </b>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c9">t</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D7" target="_self" title="REWRITE2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E5:82_1_1_1_3_1_1"/>
 ex <font color="Olive" title="b1">q</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_3_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c3">T</font>
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E66">A14</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#T33" target="_self" title="REWRITE2:th.33">Th33</a></i>;<br/><a NAME="E3:82_1_1_1_3_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E10:82_1_1_1_3"><i><font color="Green" title="E65">A13</font></i></a></i>;<br/><a NAME="E4:82_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:82"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="rewrite2.html#T46" target="_self" title="REWRITE2:th.46">Th46</a></i>;<br/><a NAME="E5:82_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:82_1_1_1_3_1_2"><i><font color="Green" title="E66">A14</font></i></a>, <a class="ref" href="rewrite2.html#T35" target="_self" title="REWRITE2:th.35">Th35</a></i>;<br/><a NAME="E6:82_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">T</font>
 ;<br/><a NAME="E7:82_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:82"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="rewrite2.html#T46" target="_self" title="REWRITE2:th.46">Th46</a></i>;<br/><a NAME="E8:82_1_1_1_3_1_2"/><b>then </b>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c9">t</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D7" target="_self" title="REWRITE2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E9:82_1_1_1_3_1_2"/>
 ex <font color="Olive" title="b1">q</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:82"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="rewrite2.html#E5:82"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="rewrite2.html#E6:82"><i><font color="Green" title="E57">A5</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c7">q</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E9:82"/><i><font color="Green" title="E58">A15</font></i>: 
( <font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c7">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E4:82"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:82"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E9:82"><i><font color="Green" title="E58">A15</font></i></a>, <a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></i>;<br/>


</div>
