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<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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1"/>
( <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1"><b>suppose </b></a><a NAME="E1:95_1_1"/><i><font color="Green" title="E70">A1</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c2">E</font> being  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/><b>take </b>
<font color="Maroon" title="c2">E</font>
;<br/><b>thus </b><a NAME="E3:95_1_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E4:95_1_1"/>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> iff <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:95_1_1"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="rewrite1.html#T27" target="_self" title="REWRITE1:th.27">Th27</a>, <a class="ref" href="rewrite1.html#T28" target="_self" title="REWRITE1:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E5:95_1_1"/>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> iff <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c2">E</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T39" target="_self" title="REWRITE1:th.39">Th39</a>, <a class="ref" href="rewrite1.html#T40" target="_self" title="REWRITE1:th.40">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2"><b>suppose </b></a><a NAME="E1:95_1_2"/>
<font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">R'</font> = <font color="Maroon" title="c1">R</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> ;<br/><b>consider </b><font color="Maroon" title="c3">xx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">R'</font>;<br/><b>consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:95_1_2"/><i><font color="Green" title="E70">A2</font></i>: 
<font color="Maroon" title="c3">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1,$2 <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>;<br/><a NAME="E6:95_1_2"/><i><font color="Green" title="E71">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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">x</font>,<font color="Olive" title="b1">x</font>]
 <b>by </b><i><a class="ref" href="rewrite1.html#T27" target="_self" title="REWRITE1:th.27">Th27</a></i>;<br/><a NAME="E7:95_1_2"/><i><font color="Green" title="E72">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>]
 <b>by </b><i><a class="txt" href="rewrite1.html#E34"><i><font color="Green" title="E28">Lm5</font></i></a></i>;<br/><a NAME="E8:95_1_2"/><i><font color="Green" title="E73">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 <b>by </b><i><a class="ref" href="rewrite1.html#T31" target="_self" title="REWRITE1:th.31">Th31</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">Q</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E10:95_1_2"/><i><font color="Green" title="E74">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c6">Q</font> iff ( <font color="Olive" title="b1">x</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="Olive" title="b2">y</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">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1" title="EQREL_1:sch.1">EQREL_1:sch 1</a>(<a class="txt" href="rewrite1.html#E6:95_1_2"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="rewrite1.html#E7:95_1_2"><i><font color="Green" title="E72">A4</font></i></a>, <a class="txt" href="rewrite1.html#E8:95_1_2"><i><font color="Green" title="E73">A5</font></i></a>);<br/></i><a NAME="E11:95_1_2"/><i><font color="Green" title="E75">A7</font></i>: 
<font color="Maroon" title="c4">x1</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#E5:95_1_2"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E12:95_1_2"/><i><font color="Green" title="E76">A8</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">Q</font> holds <br/><font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="rewrite1.html#E11:95_1_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E13:95_1_2"/><i><font color="Green" title="E77">A9</font></i>: 
for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">Q</font> &amp; <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">Q</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">Y</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E11:95_1_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E14:95_1_2"/>
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E11:95_1_2"><i><font color="Green" title="E75">A7</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:95_1_2"/><i><font color="Green" title="E78">A10</font></i>: 
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="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">Q</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c7">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rewrite1.html#E12:95_1_2"><i><font color="Green" title="E76">A8</font></i></a>, <a class="txt" href="rewrite1.html#E13:95_1_2"><i><font color="Green" title="E77">A9</font></i></a>, <a class="ref" href="wellord2.html#T27" title="WELLORD2:th.27">WELLORD2:27</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $2 &amp; $1,$2 <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> );<br/><b>consider </b><font color="Maroon" title="c8">P</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E18:95_1_2"/><i><font color="Green" title="E79">A11</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c8">P</font> iff ( <font color="Olive" title="b1">x</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="Olive" title="b2">y</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">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1" title="RELAT_1:sch.1">RELAT_1:sch 1</a>();<br/></i><a NAME="E19:95_1_2"/><i><font color="Green" title="E80">A12</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c8">P</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>given </b><font color="Maroon" title="c11">p</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of <font color="Maroon" title="c8">P</font><b> such that </b><a NAME="E1:95_1_2_1"/><i><font color="Green" title="E81">A13</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c11">p</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D3" target="_self" title="REWRITE1:def.3">REWRITE1:def 3</a><br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $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="c11">p</font> implies <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> );<br/>
<a NAME="E2:95_1_2_1"/><i><font color="Green" title="E82">A14</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="rewrite1.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:95_1_2_1_1"/><i><font color="Green" title="E83">A15</font></i>: 
( <font color="Maroon" title="c12">i</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="c11">p</font> implies <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> )
 <b>and </b><br/><a NAME="E2:95_1_2_1_1"/><i><font color="Green" title="E84">A16</font></i>: 
<font color="Maroon" title="c12">i</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="c11">p</font>
 ;<br/><a NAME="E3:95_1_2_1_1"/><i><font color="Green" title="E85">A17</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E2:95_1_2_1_1"><i><font color="Green" title="E84">A16</font></i></a>, <a class="txt" href="rewrite1.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_1_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_1_1_1_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_1_1_1_1"/>
<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:95_1_2_1"><i><font color="Green" title="E81">A13</font></i></a>, <a class="ref" href="rewrite1.html#T27" target="_self" title="REWRITE1:th.27">Th27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_1_1_1_2"/><i><font color="Green" title="E86">A18</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:95_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">i</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="c11">p</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E3:95_1_2_1_1"><i><font color="Green" title="E85">A17</font></i></a>, <a class="txt" href="rewrite1.html#E4"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/><a NAME="E3:95_1_2_1_1_1_2"/><b>then </b>
<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="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i</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> <font color="Maroon" title="c8">P</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E2:95_1_2_1_1"><i><font color="Green" title="E84">A16</font></i></a>, <a class="ref" href="rewrite1.html#D2" target="_self" title="REWRITE1:def.2">Def2</a></i>;<br/><a NAME="E4:95_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E18:95_1_2"><i><font color="Green" title="E79">A11</font></i></a>, <a class="txt" href="rewrite1.html#E1:95_1_2_1_1"><i><font color="Green" title="E83">A15</font></i></a>, <a class="txt" href="rewrite1.html#E3:95_1_2_1_1"><i><font color="Green" title="E85">A17</font></i></a>, <a class="txt" href="rewrite1.html#E1:95_1_2_1_1_1_2"><i><font color="Green" title="E86">A18</font></i></a>, <a class="txt" href="rewrite1.html#E4"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:95_1_2_1_1_1_2"/>
<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T31" target="_self" title="REWRITE1:th.31">Th31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:95_1_2_1"/><b>then </b><i><font color="Green" title="E83">A19</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<a NAME="E5:95_1_2_1"/><i><font color="Green" title="E84">A20</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="rewrite1.html#E2:95_1_2_1"><i><font color="Green" title="E82">A14</font></i></a>, <a class="txt" href="rewrite1.html#E4:95_1_2_1"><i><font color="Green" title="E83">A19</font></i></a>);<br/></i>
<a NAME="E6:95_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">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="c11">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/>
<b>hence </b><a NAME="E7:95_1_2_1"/>
<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:95_1_2_1"><i><font color="Green" title="E81">A13</font></i></a>, <a class="txt" href="rewrite1.html#E5:95_1_2_1"><i><font color="Green" title="E84">A20</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:95_1_2"/><i><font color="Green" title="E81">A21</font></i>: 
<font color="Maroon" title="c8">P</font> is <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D15" target="_self" title="REWRITE1:def.15">REWRITE1:def 15</a><br/>

<b>consider </b><font color="Maroon" title="c10">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_2_1"/>
(  not <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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</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> <font color="Maroon" title="c8">P</font> or <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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</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> <font color="Maroon" title="c8">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_2_1_1"/>
not <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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</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> <font color="Maroon" title="c8">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_2_1_1"/>
not for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</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> <font color="Maroon" title="c8">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_2_1_2"/><i><font color="Green" title="E82">A22</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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</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> <font color="Maroon" title="c8">P</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">j</font> = <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>assume </b><a NAME="E2:95_1_2_2_1_2"/><i><font color="Green" title="E83">A23</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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">j</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> <font color="Maroon" title="c8">P</font>
 ;<br/><a NAME="E3:95_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E84">A24</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E18:95_1_2"><i><font color="Green" title="E79">A11</font></i></a>, <a class="txt" href="rewrite1.html#E1:95_1_2_2_1_2"><i><font color="Green" title="E82">A22</font></i></a></i>;<br/><a NAME="E4:95_1_2_2_1_2"/><b>then </b>
<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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">j</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> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E10:95_1_2"><i><font color="Green" title="E74">A6</font></i></a></i>;<br/><a NAME="E5:95_1_2_2_1_2"/><b>then </b>
<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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></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="c6">Q</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/><a NAME="E6:95_1_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E3:95_1_2_2_1_2"><i><font color="Green" title="E84">A24</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/><a NAME="E7:95_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E85">A25</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span> &amp;  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">Q</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E3:95_1_2_2_1_2"><i><font color="Green" title="E84">A24</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:95_1_2_2_1_2"/><i><font color="Green" title="E86">A26</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rewrite1.html#E16:95_1_2"><i><font color="Green" title="E78">A10</font></i></a></i>;<br/><a NAME="E10:95_1_2_2_1_2"/>
( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E7:95_1_2_2_1_2"><i><font color="Green" title="E85">A25</font></i></a>, <a class="txt" href="rewrite1.html#E9:95_1_2_2_1_2"><i><font color="Green" title="E86">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E11:95_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rewrite1.html#E18:95_1_2"><i><font color="Green" title="E79">A11</font></i></a>, <a class="txt" href="rewrite1.html#E2:95_1_2_2_1_2"><i><font color="Green" title="E83">A23</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E21:95_1_2"/>
<font color="Maroon" title="c8">P</font> is <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c11">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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>assume </b><a NAME="E1:95_1_2_3"/><i><font color="Green" title="E82">A27</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="c8">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">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="c8">P</font> )
 ;<br/>

<b>take </b>
<font color="Maroon" title="c10">b</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">REWRITE1:def 7</a><br/>

<a NAME="E2:95_1_2_3"/><i><font color="Green" title="E83">A28</font></i>: 
( <font color="Maroon" title="c9">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="c10">b</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="c11">c</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="c9">a</font>,<font color="Maroon" title="c10">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">c</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E18:95_1_2"><i><font color="Green" title="E79">A11</font></i></a>, <a class="txt" href="rewrite1.html#E1:95_1_2_3"><i><font color="Green" title="E82">A27</font></i></a></i>;<br/>
<a NAME="E3:95_1_2_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="c6">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">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="c6">Q</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E10:95_1_2"><i><font color="Green" title="E74">A6</font></i></a></i>;<br/>
<a NAME="E4:95_1_2_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c9">a</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="c6">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c9">a</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="c6">Q</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/>
<a NAME="E5:95_1_2_3"/><b>then </b><i><font color="Green" title="E84">A29</font></i>: 
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c9">a</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<a NAME="E6:95_1_2_3"/>
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">Q</font>
 
<b>by </b><i><a class="txt" href="rewrite1.html#E2:95_1_2_3"><i><font color="Green" title="E83">A28</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:95_1_2_3"/><i><font color="Green" title="E85">A30</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rewrite1.html#E16:95_1_2"><i><font color="Green" title="E78">A10</font></i></a></i>;<br/>
<a NAME="E9:95_1_2_3"/>
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E2:95_1_2_3"><i><font color="Green" title="E83">A28</font></i></a>, <a class="txt" href="rewrite1.html#E5:95_1_2_3"><i><font color="Green" title="E84">A29</font></i></a>, <a class="txt" href="rewrite1.html#E8:95_1_2_3"><i><font color="Green" title="E85">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:95_1_2_3"/><b>then </b>
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:95_1_2_3"/>
( <font color="Maroon" title="c8">P</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c8">P</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T13" target="_self" title="REWRITE1:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">P</font> = <font color="Maroon" title="c8">P</font> as  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a> <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a> <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> <b>by </b><i><a class="txt" href="rewrite1.html#E20:95_1_2"><i><font color="Green" title="E81">A21</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c9">P</font>
;<br/><b>thus </b><a NAME="E23:95_1_2"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1_2_4"/>
<font color="Maroon" title="c10">x</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="c9">P</font>
 ;<br/>

<a NAME="E2:95_1_2_4"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">P</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">P</font></span>)</span>
 
;<br/>
<a NAME="E3:95_1_2_4"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">P</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">P</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:95_1_2_4"/><b>then </b>
(  ex <font color="Olive" title="b1">y</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="c10">x</font>,<font color="Olive" title="b1">y</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="c9">P</font> or  ex <font color="Olive" title="b1">y</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">y</font>,<font color="Maroon" title="c10">x</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="c9">P</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:95_1_2_4"/>
<font color="Maroon" title="c10">x</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#E18:95_1_2"><i><font color="Green" title="E79">A11</font></i></a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E24:95_1_2"/>
( <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> iff <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c9">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_2_5_1"/>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> or <font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:95_1_2_5_1_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:95_1_2_5_1_1"/>
( <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> iff <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c9">P</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T27" target="_self" title="REWRITE1:th.27">Th27</a>, <a class="ref" href="rewrite1.html#T39" target="_self" title="REWRITE1:th.39">Th39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:95_1_2_5_1_2"/><i><font color="Green" title="E82">A31</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b</font>
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:95_1_2_5_1_2_1"/><i><font color="Green" title="E83">A32</font></i>: 
<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E2:95_1_2_5_1_2_1"/><b>then </b><i><font color="Green" title="E84">A33</font></i>: 
( <font color="Maroon" title="c10">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="c11">b</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#E1:95_1_2_5_1_2"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="rewrite1.html#T33" target="_self" title="REWRITE1:th.33">Th33</a></i>;<br/><a NAME="E3:95_1_2_5_1_2_1"/><b>then </b><i><font color="Green" title="E85">A34</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">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="c6">Q</font> &amp;  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">Q</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c11">b</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E10:95_1_2"><i><font color="Green" title="E74">A6</font></i></a>, <a class="txt" href="rewrite1.html#E1:95_1_2_5_1_2_1"><i><font color="Green" title="E83">A32</font></i></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:95_1_2_5_1_2_1"/><i><font color="Green" title="E86">A35</font></i>: 
<font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c11">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rewrite1.html#E16:95_1_2"><i><font color="Green" title="E78">A10</font></i></a></i>;<br/><b>thus </b><a NAME="E6:95_1_2_5_1_2_1"/>
<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c9">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2_5_1_2_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c12">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">REWRITE1:def 7</a><br/>

<a NAME="E1:95_1_2_5_1_2_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:95_1_2_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E87">A36</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c11">b</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E5:95_1_2_5_1_2_1"><i><font color="Green" title="E86">A35</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:95_1_2_5_1_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">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="c6">Q</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c11">b</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E3:95_1_2_5_1_2_1"><i><font color="Green" title="E85">A34</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<a NAME="E4:95_1_2_5_1_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">x</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="c6">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">x</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="c6">Q</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E2:95_1_2_5_1_2_1_1"><i><font color="Green" title="E87">A36</font></i></a>, <a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a>, <a class="ref" href="eqrel_1.html#T30" title="EQREL_1:th.30">EQREL_1:30</a></i>;<br/>
<a NAME="E5:95_1_2_5_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c12">x</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="c11">b</font>,<font color="Maroon" title="c12">x</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">x</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E10:95_1_2"><i><font color="Green" title="E74">A6</font></i></a></i>;<br/>
<a NAME="E6:95_1_2_5_1_2_1_1"/><b>then </b>
( ( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">x</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="c9">P</font> ) &amp; ( <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">x</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="c9">P</font> ) )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E18:95_1_2"><i><font color="Green" title="E79">A11</font></i></a>, <a class="txt" href="rewrite1.html#E2:95_1_2_5_1_2_1"><i><font color="Green" title="E84">A33</font></i></a>, <a class="txt" href="rewrite1.html#E2:95_1_2_5_1_2_1_1"><i><font color="Green" title="E87">A36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:95_1_2_5_1_2_1_1"/>
( <font color="Maroon" title="c9">P</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c9">P</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">x</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T13" target="_self" title="REWRITE1:th.13">Th13</a>, <a class="ref" href="rewrite1.html#T16" target="_self" title="REWRITE1:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div><b>given </b><font color="Maroon" title="c12">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:95_1_2_5_1_2"/><i><font color="Green" title="E83">A37</font></i>: 
( <font color="Maroon" title="c9">P</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">c</font> &amp; <font color="Maroon" title="c9">P</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">REWRITE1:def 7</a><br/><a NAME="E4:95_1_2_5_1_2"/>
<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E19:95_1_2"><i><font color="Green" title="E80">A12</font></i></a>, <a class="txt" href="rewrite1.html#E3:95_1_2_5_1_2"><i><font color="Green" title="E83">A37</font></i></a></i>;<br/><a NAME="E5:95_1_2_5_1_2"/><b>then </b>
( <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">c</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c12">c</font>,<font color="Maroon" title="c11">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E19:95_1_2"><i><font color="Green" title="E80">A12</font></i></a>, <a class="txt" href="rewrite1.html#E3:95_1_2_5_1_2"><i><font color="Green" title="E83">A37</font></i></a>, <a class="txt" href="rewrite1.html#E34"><i><font color="Green" title="E28">Lm5</font></i></a></i>;<br/><b>hence </b><a NAME="E6:95_1_2_5_1_2"/>
<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T31" target="_self" title="REWRITE1:th.31">Th31</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>
