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

<b>reconsider </b><font color="Maroon">R</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T4">RELAT_1:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

<a NAME="E2:86_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T32">RELAT_1:32</a></i>;<br/>
<b>thus </b><a NAME="E3:86_1_1"/>
<font color="Maroon">R</font> is <a href="rewrite1.html#V7" target="_self">confluent</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D25" target="_self">REWRITE1:def 25</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D22" target="_self">REWRITE1:def 22</a><br/>

<b>given </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:86_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D8" target="_self">REWRITE1:def 8</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_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="E1:86_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:86_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:86_1_1_1_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:86_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:86_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E3:86_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T14" target="_self">Th14</a>, <a class="txt" href="rewrite1.html#E2"><i><font color="Green">Lemma22</font></i></a></i>;<br/><a NAME="E4:86_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T13" target="_self">Th13</a>, <a class="ref" href="rewrite1.html#T14" target="_self">Th14</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:86_1_1_1_1_2"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 ) &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> 0,1 &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> 1,1 )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:86_1_1_1_1_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<a NAME="E4:86_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> is <a href="relat_2.html#V2">irreflexive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D2">RELAT_2:def 2</a>,<a class="ref" href="relat_2.html#D10">RELAT_2:def 10</a><br/>

<b>assume </b><a NAME="E1:86_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E2:86_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:86_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E4:86_1_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:86_1_1"/>
<font color="Maroon">R</font> is <a href="rewrite1.html#V1" target="_self">co-well_founded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D16" target="_self">REWRITE1:def 16</a><br/>

<b>assume </b><a NAME="E1:86_1_1_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y'</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_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="E1:86_1_1_3_1"/>
( 1 <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or ( not 1 <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:86_1_1_3_1_1"/><i><font color="Green">E33</font></i>: 
1 <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:86_1_1_3_1_1"/>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T16" target="_self">Th16</a></i>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:86_1_1_3_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E4:86_1_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:86_1_1_3_1_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:86_1_1_3_1_2"/><i><font color="Green">E35</font></i>: 
( not 1 <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><div class="add"><b>take </b>
0
;<br/><b>thus </b><a NAME="E2:86_1_1_3_1_2"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T13" target="_self">Th13</a>, <a class="ref" href="rewrite1.html#T15" target="_self">Th15</a>, <a class="ref" href="rewrite1.html#T16" target="_self">Th16</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:86_1_1_3_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>hence </b><a NAME="E4:86_1_1_3_1_2"/>
(  not 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> or  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T13" target="_self">Th13</a>, <a class="ref" href="rewrite1.html#T15" target="_self">Th15</a>, <a class="ref" href="rewrite1.html#T16" target="_self">Th16</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:86_1_1"/><b>then </b>
<font color="Maroon">R</font> is  <a href="relat_2.html#V2">irreflexive</a> <a href="rewrite1.html#V1" target="_self">co-well_founded</a> <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i><a class="ref" href="rewrite1.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E7:86_1_1"/>
<font color="Maroon">R</font> is <a href="rewrite1.html#V3" target="_self">strongly-normalizing</a>
 ;<br/>


</div>
