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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a>  &amp; <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>  implies <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:80_1_1"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>  or <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>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/>
<span class="kw">assume </span><a NAME="E2:80_1_1"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
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>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D24" target="_self" title="REWRITE1:def.24">REWRITE1:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a> </span><br/>

<a NAME="E3:80_1_1"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
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="80_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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="c2">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="c2">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>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:80_1_1_1"/><span class="lab"><font color="Green" title="E69">A4</font></span>: 
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="c2">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="c2">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>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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="c2">a</font>,<font color="Maroon" title="c4">c</font> implies <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:80_1_1_1"/><span class="lab"><font color="Green" title="E70">A5</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E3:80_1_1_1"/><span class="lab"><font color="Green" title="E71">A6</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E5:80_1_1_1"/><span class="lab"><font color="Green" title="E72">A7</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E6:80_1_1_1"/><span class="lab"><font color="Green" title="E73">A8</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c5">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_1"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="ref" href="rewrite1.html#D3" target="_self" title="REWRITE1:def.3">Def3</a></span>;<br/>
<a NAME="E7:80_1_1_1"/><span class="lab"><font color="Green" title="E74">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">q</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E9:80_1_1_1"/><span class="lab"><font color="Green" title="E75">A10</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E10:80_1_1_1"/><span class="lab"><font color="Green" title="E76">A11</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c6">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1_1_1"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="ref" href="rewrite1.html#D3" target="_self" title="REWRITE1:def.3">Def3</a></span>;<br/>
<a NAME="E11:80_1_1_1"/><span class="lab"><font color="Green" title="E77">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="c6">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="c5">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="c6">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="80_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:80_1_1_1_1_1"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1_1_1"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="txt" href="#E5:80_1_1_1"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="txt" href="#E6:80_1_1_1"><span class="lab"><font color="Green" title="E73">A8</font></span></a>, <a class="ref" href="rewrite1.html#T37" target="_self" title="REWRITE1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:80_1_1_1_1_2"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1_1"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="txt" href="#E9:80_1_1_1"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="txt" href="#E10:80_1_1_1"><span class="lab"><font color="Green" title="E76">A11</font></span></a>, <a class="ref" href="rewrite1.html#T37" target="_self" title="REWRITE1:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:80_1_1_1_1_3"/><span class="lab"><font color="Green" title="E78">A13</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="c6">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font></span><br/><div class="add"><a NAME="E2:80_1_1_1_1_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80_1_1_1"><span class="lab"><font color="Green" title="E77">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A14</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A15</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="rewrite1.html#E4"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><a NAME="E5:80_1_1_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/><a NAME="E6:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A16</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:80_1_1_1"><span class="lab"><font color="Green" title="E76">A11</font></span></a>, <a class="txt" href="#E3:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E79">A14</font></span></a>, <a class="txt" href="#E4:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="ref" href="rewrite1.html#T18" target="_self" title="REWRITE1:th.18">Th18</a></span>;<br/><a NAME="E7:80_1_1_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_1_1"><span class="lab"><font color="Green" title="E74">A9</font></span></a>, <a class="txt" href="#E1:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E78">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E8:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A17</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A18</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="rewrite1.html#E4"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><a NAME="E10:80_1_1_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/><a NAME="E11:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A19</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:80_1_1_1"><span class="lab"><font color="Green" title="E73">A8</font></span></a>, <a class="txt" href="#E8:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E82">A17</font></span></a>, <a class="txt" href="#E9:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A18</font></span></a>, <a class="ref" href="rewrite1.html#T18" target="_self" title="REWRITE1:th.18">Th18</a></span>;<br/><a NAME="E12:80_1_1_1_1_3"/>
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="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_1_1"><span class="lab"><font color="Green" title="E74">A9</font></span></a>, <a class="txt" href="rewrite1.html#E4"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><a NAME="E13:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_1_1_1"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="txt" href="#E9:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A18</font></span></a>, <a class="ref" href="rewrite1.html#D2" target="_self" title="REWRITE1:def.2">Def2</a></span>;<br/><a NAME="E14:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A21</font></span>: 
<font color="Maroon" title="c2">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></span>;<br/><a NAME="E15:80_1_1_1_1_3"/>
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="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80_1_1_1"><span class="lab"><font color="Green" title="E77">A12</font></span></a>, <a class="txt" href="rewrite1.html#E4"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><a NAME="E16:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A22</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_1_1_1"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="txt" href="#E4:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E80">A15</font></span></a>, <a class="ref" href="rewrite1.html#D2" target="_self" title="REWRITE1:def.2">Def2</a></span>;<br/><a NAME="E17:80_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_1"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E13:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E85">A20</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:80_1_1_1_1_3"/><span class="lab"><font color="Green" title="E88">A23</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c7">d</font>
 <span class="kw">and </span><br/><a NAME="E20:80_1_1_1_1_3"/><span class="lab"><font color="Green" title="E89">A24</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></span>;<br/><a NAME="E21:80_1_1_1_1_3"/><span class="lab"><font color="Green" title="E90">A25</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_1"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="relat_2.html#D10" title="RELAT_2:def.10">RELAT_2:def 10</a></span>;<br/><a NAME="E22:80_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A26</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E87">A22</font></span></a>, <a class="txt" href="#E14:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A21</font></span></a>, <a class="ref" href="relat_2.html#D2" title="RELAT_2:def.2">RELAT_2:def 2</a></span>;<br/><a NAME="E23:80_1_1_1_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E14:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A21</font></span></a>, <a class="txt" href="#E21:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E90">A25</font></span></a>, <a class="ref" href="relat_2.html#D2" title="RELAT_2:def.2">RELAT_2:def 2</a></span>;<br/><a NAME="E24:80_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">d</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_1_1"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="txt" href="#E13:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E19:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E88">A23</font></span></a>, <a class="txt" href="#E11:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A19</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E26:80_1_1_1_1_3"/><span class="lab"><font color="Green" title="E92">A27</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">e</font>
 <span class="kw">and </span><br/><a NAME="E27:80_1_1_1_1_3"/><span class="lab"><font color="Green" title="E93">A28</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></span>;<br/><a NAME="E28:80_1_1_1_1_3"/>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2,<font color="Maroon" title="c8">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E89">A24</font></span></a>, <a class="txt" href="#E27:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E93">A28</font></span></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></span>;<br/><a NAME="E29:80_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_1_1"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="txt" href="#E16:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E87">A22</font></span></a>, <a class="txt" href="#E22:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E91">A26</font></span></a>, <a class="txt" href="#E6:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E81">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E30:80_1_1_1_1_3"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:80_1_1_1_1_3"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="ref" href="rewrite1.html#T43" target="_self" title="REWRITE1:th.43">Th43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:80_1_1"/><span class="lab"><font color="Green" title="E69">A29</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V1" title="REWRITE1:attr.1">co-well_founded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_1"><span class="lab"><font color="Green" title="E66">A1</font></span></a></span>;<br/>
<a NAME="E5:80_1_1"/><span class="lab"><font color="Green" title="E70">A30</font></span>: 
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="rewrite1.html#S2" target="_self" title="REWRITE1:sch.2">REWRITE1:sch 2</a>(<a class="txt" href="#E4:80_1_1"><span class="lab"><font color="Green" title="E69">A29</font></span></a>, <a class="txt" href="#E3:80_1_1"><span class="lab"><font color="Green" title="E68">A3</font></span></a>);<br/></span>
<span class="kw">given </span><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> <span class="kw"> such that </span><a NAME="E7:80_1_1"/><span class="lab"><font color="Green" title="E71">A31</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:80_1_1"/><span class="lab"><font color="Green" title="E72">A32</font></span>: 
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>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D22" target="_self" title="REWRITE1:def.22">REWRITE1:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">c0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:80_1_1"/><span class="lab"><font color="Green" title="E73">A33</font></span>: 
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_1"><span class="lab"><font color="Green" title="E71">A31</font></span></a>, <a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">Def8</a></span>;<br/>
<a NAME="E11:80_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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_1_1"><span class="lab"><font color="Green" title="E72">A32</font></span></a>, <a class="ref" href="rewrite1.html#T39" target="_self" title="REWRITE1:th.39">Th39</a></span>;<br/>
<a NAME="E12:80_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:80_1_1"><span class="lab"><font color="Green" title="E73">A33</font></span></a>, <a class="ref" href="rewrite1.html#T19" target="_self" title="REWRITE1:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:80_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_1_1"><span class="lab"><font color="Green" title="E72">A32</font></span></a>, <a class="txt" href="#E10:80_1_1"><span class="lab"><font color="Green" title="E73">A33</font></span></a>, <a class="txt" href="#E5:80_1_1"><span class="lab"><font color="Green" title="E70">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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