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

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:82_1_1"/>
<font color="Maroon" title="c1">R</font> is <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a>
 ;<br/>

<a NAME="E2:82_1_1"/><b>then </b><i><font color="Green" title="E66">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a> <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 
;<br/>
<a NAME="E3:82_1_1"/><b>then </b><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="rewrite1.html#V1" title="REWRITE1:attr.1">co-well_founded</a>
 
;<br/>
<b>assume </b><a NAME="E4:82_1_1"/><i><font color="Green" title="E68">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D24" target="_self" title="REWRITE1:def.24">REWRITE1:def 24</a><br/>

<b>given </b><font color="Maroon" title="c2">a0</font>, <font color="Maroon" title="c3">b0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E6:82_1_1"/><i><font color="Green" title="E69">A4</font></i>: 
( <font color="Maroon" title="c2">a0</font>,<font color="Maroon" title="c3">b0</font> <a href="rewrite1.html#R6" title="REWRITE1:pred.6">are_divergent_wrt</a> <font color="Maroon" title="c1">R</font> &amp; not <font color="Maroon" title="c2">a0</font>,<font color="Maroon" title="c3">b0</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D22" target="_self" title="REWRITE1:def.22">REWRITE1:def 22</a><br/>

<b>consider </b><font color="Maroon" title="c4">c0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:82_1_1"/><i><font color="Green" title="E70">A5</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c0</font>,<font color="Maroon" title="c2">a0</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c0</font>,<font color="Maroon" title="c3">b0</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E6:82_1_1"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">Def8</a></i>;<br/>
<a NAME="E9:82_1_1"/>
( <font color="Maroon" title="c2">a0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c0</font> or <font color="Maroon" title="c3">b0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c0</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E6:82_1_1"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="rewrite1.html#T39" target="_self" title="REWRITE1:th.39">Th39</a></i>;<br/>
<a NAME="E10:82_1_1"/><b>then </b><i><font color="Green" title="E71">A6</font></i>: 
<font color="Maroon" title="c4">c0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="rewrite1.html#E8:82_1_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="ref" href="rewrite1.html#T19" target="_self" title="REWRITE1:th.19">Th19</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive" title="b2">c</font> holds <br/><font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>;<br/>
<a NAME="E11:82_1_1"/><i><font color="Green" title="E72">A7</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( for <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:82_1_1_1"/><i><font color="Green" title="E73">A8</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:82_1_1_1"/><i><font color="Green" title="E74">A9</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:82_1_1_1"/><i><font color="Green" title="E75">A10</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3" target="_self" title="REWRITE1:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">q</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E6:82_1_1_1"/><i><font color="Green" title="E76">A11</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="c9">q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E2:82_1_1_1"><i><font color="Green" title="E74">A9</font></i></a>, <a class="ref" href="rewrite1.html#D3" target="_self" title="REWRITE1:def.3">Def3</a></i>;<br/>
<a NAME="E7:82_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="rewrite1.html#D2" target="_self" title="REWRITE1:def.2">Def2</a></i>;<br/>
<a NAME="E8:82_1_1_1"/><b>then </b><i><font color="Green" title="E77">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_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_1"/>
(  <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> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_1_1"/>
 <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> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:82_1_1_1_1_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E2:82_1_1_1"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="rewrite1.html#E4:82_1_1_1"><i><font color="Green" title="E75">A10</font></i></a>, <a class="ref" href="rewrite1.html#T37" target="_self" title="REWRITE1:th.37">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:82_1_1_1_1_2"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E2:82_1_1_1"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="rewrite1.html#E6:82_1_1_1"><i><font color="Green" title="E76">A11</font></i></a>, <a class="ref" href="rewrite1.html#T37" target="_self" title="REWRITE1:th.37">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:82_1_1_1_1_3"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:82_1_1_1_1_3"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="rewrite1.html#E8:82_1_1_1"><i><font color="Green" title="E77">A12</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:82_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E78">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> &amp; 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:82_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E79">A14</font></i>: 
( 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> &amp; 1 <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> &amp; 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="c9">q</font> &amp; 1 <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="c9">q</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E8:82_1_1_1"><i><font color="Green" title="E77">A12</font></i></a>, <a class="txt" href="rewrite1.html#E4"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/><a NAME="E5:82_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E80">A15</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<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> 2</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> <font color="Maroon" title="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</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> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E4:82_1_1_1"><i><font color="Green" title="E75">A10</font></i></a>, <a class="txt" href="rewrite1.html#E6:82_1_1_1"><i><font color="Green" title="E76">A11</font></i></a>, <a class="ref" href="rewrite1.html#D2" target="_self" title="REWRITE1:def.2">Def2</a></i>;<br/><a NAME="E6:82_1_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c1">R</font> <a href="relat_2.html#R2" title="RELAT_2:pred.2">is_irreflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E2:82_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a>, <a class="ref" href="relat_2.html#D10" title="RELAT_2:def.10">RELAT_2:def 10</a></i>;<br/><a NAME="E7:82_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E81">A16</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 <b>by </b><i><a class="txt" href="rewrite1.html#E5:82_1_1_1_1_3"><i><font color="Green" title="E80">A15</font></i></a>, <a class="ref" href="relat_2.html#D2" title="RELAT_2:def.2">RELAT_2:def 2</a></i>;<br/><a NAME="E8:82_1_1_1_1_3"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E4:82_1_1"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="rewrite1.html#E5:82_1_1_1_1_3"><i><font color="Green" title="E80">A15</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:82_1_1_1_1_3"/><i><font color="Green" title="E82">A17</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c10">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c10">d</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></i>;<br/><a NAME="E11:82_1_1_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</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>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E12:82_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E4:82_1_1_1"><i><font color="Green" title="E75">A10</font></i></a>, <a class="txt" href="rewrite1.html#E3:82_1_1_1_1_3"><i><font color="Green" title="E78">A13</font></i></a>, <a class="txt" href="rewrite1.html#E4:82_1_1_1_1_3"><i><font color="Green" title="E79">A14</font></i></a>, <a class="ref" href="rewrite1.html#T18" target="_self" title="REWRITE1:th.18">Th18</a></i>;<br/><a NAME="E13:82_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">d</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:82_1_1_1"><i><font color="Green" title="E73">A8</font></i></a>, <a class="txt" href="rewrite1.html#E5:82_1_1_1_1_3"><i><font color="Green" title="E80">A15</font></i></a>, <a class="txt" href="rewrite1.html#E7:82_1_1_1_1_3"><i><font color="Green" title="E81">A16</font></i></a>, <a class="txt" href="rewrite1.html#E10:82_1_1_1_1_3"><i><font color="Green" title="E82">A17</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:82_1_1_1_1_3"/><i><font color="Green" title="E83">A18</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c11">e</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c11">e</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></i>;<br/><a NAME="E16:82_1_1_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</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="c9">q</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E17:82_1_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c11">e</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E6:82_1_1_1"><i><font color="Green" title="E76">A11</font></i></a>, <a class="txt" href="rewrite1.html#E3:82_1_1_1_1_3"><i><font color="Green" title="E78">A13</font></i></a>, <a class="txt" href="rewrite1.html#E4:82_1_1_1_1_3"><i><font color="Green" title="E79">A14</font></i></a>, <a class="txt" href="rewrite1.html#E10:82_1_1_1_1_3"><i><font color="Green" title="E82">A17</font></i></a>, <a class="txt" href="rewrite1.html#E15:82_1_1_1_1_3"><i><font color="Green" title="E83">A18</font></i></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a>, <a class="ref" href="rewrite1.html#T18" target="_self" title="REWRITE1:th.18">Th18</a></i>;<br/><a NAME="E18:82_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c7">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:82_1_1_1"><i><font color="Green" title="E73">A8</font></i></a>, <a class="txt" href="rewrite1.html#E5:82_1_1_1_1_3"><i><font color="Green" title="E80">A15</font></i></a>, <a class="txt" href="rewrite1.html#E7:82_1_1_1_1_3"><i><font color="Green" title="E81">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E19:82_1_1_1_1_3"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E15:82_1_1_1_1_3"><i><font color="Green" title="E83">A18</font></i></a>, <a class="ref" href="rewrite1.html#T43" target="_self" title="REWRITE1:th.43">Th43</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:82_1_1"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 
<b>from </b><i><a class="ref" href="rewrite1.html#S2" target="_self" title="REWRITE1:sch.2">REWRITE1:sch 2</a>(<a class="txt" href="rewrite1.html#E3:82_1_1"><i><font color="Green" title="E67">A2</font></i></a>, <a class="txt" href="rewrite1.html#E11:82_1_1"><i><font color="Green" title="E72">A7</font></i></a>);<br/></i>
<b>hence </b><a NAME="E13:82_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rewrite1.html#E6:82_1_1"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="rewrite1.html#E8:82_1_1"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="rewrite1.html#E10:82_1_1"><i><font color="Green" title="E71">A6</font></i></a></i>;<br/>


</div>
