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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <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> ;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E48">A1</font></i>: 
( ( <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="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> ) or ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</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">c</font>,<font color="Maroon" title="c3">b</font> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( ( <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="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> ) or ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</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">c</font>,<font color="Maroon" title="c3">b</font> ) )
 </span><b>by </b><i><a class="txt" href="rewrite1.html#E1:53"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/><i><font color="Green" title="E49">A2</font></i>: 
( <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="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> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1"/><i><font color="Green" title="E50">A3</font></i>: 
( <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="c5">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></i>;<br/><a NAME="E4:53_1_1"/>
<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="c5">d</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:53_1_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="rewrite1.html#E3:53_1_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E5:53_1_1"/>
<font color="Maroon" title="c2">a</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>
 <b>by </b><i><a class="txt" href="rewrite1.html#E3:53_1_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/><i><font color="Green" title="E49">A4</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</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">c</font>,<font color="Maroon" title="c3">b</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:53_1_2"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></i>;<br/><a NAME="E4:53_1_2"/>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:53_1_2"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="rewrite1.html#E3:53_1_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E5:53_1_2"/>
<font color="Maroon" title="c2">a</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>
 <b>by </b><i><a class="txt" href="rewrite1.html#E3:53_1_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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