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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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:89_1"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a>
 ;<br/><b>thus </b><a NAME="E2:89_1"/>
<font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  is <a href="rewrite1.html#V6" title="REWRITE1:attr.6">subcommutative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>proof </b></a><div class="add">

<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D21" target="_self" title="REWRITE1:def.21">REWRITE1:def 21</a><br/>

<b>assume </b><a NAME="E1:89_1_1"/>
( <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="Maroon" title="c3">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> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  &amp; <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="Maroon" title="c4">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> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  )
 ;<br/>

<a NAME="E2:89_1_1"/><b>then </b>
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="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> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#T21" target="_self" title="REWRITE1:th.21">Th21</a></i>;<br/>
<a NAME="E3:89_1_1"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R6" title="REWRITE1:pred.6">are_divergent_wrt</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">Def8</a></i>;<br/>
<a NAME="E4:89_1_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="rewrite1.html#E1:89_1"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="rewrite1.html#D22" target="_self" title="REWRITE1:def.22">Def22</a></i>;<br/>
<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="E6:89_1_1"/><i><font color="Green" title="E68">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="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/>
<b>take </b>
<font color="Maroon" title="c5">d</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D9" target="_self" title="REWRITE1:def.9">REWRITE1:def 9</a><br/>

<b>thus </b><a NAME="E7:89_1_1"/>
( ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</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> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</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> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> ) )
 <b>by </b><i><a class="txt" href="rewrite1.html#E6:89_1_1"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="rewrite1.html#T21" target="_self" title="REWRITE1:th.21">Th21</a></i>;<br/>


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

<b>assume </b><a NAME="E2:89"/><i><font color="Green" title="E67">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  &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> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="rewrite1.html#R7" title="REWRITE1:pred.7">are_convergent&lt;=1_wrt</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D21" target="_self" title="REWRITE1:def.21">REWRITE1:def 21</a><br/>

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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#D22" target="_self" title="REWRITE1:def.22">REWRITE1:def 22</a><br/>

<b>given </b><font color="Maroon" title="c4">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:89"/><i><font color="Green" title="E68">A4</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</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> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">REWRITE1:def 8</a><br/>

<a NAME="E4:89"/>
( ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">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="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">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> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E3:89"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="rewrite1.html#T21" target="_self" title="REWRITE1:th.21">Th21</a></i>;<br/>
<a NAME="E5:89"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R7" title="REWRITE1:pred.7">are_convergent&lt;=1_wrt</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a> 
 
<b>by </b><i><a class="txt" href="rewrite1.html#E2:89"><i><font color="Green" title="E67">A3</font></i></a>, <a class="ref" href="rewrite1.html#D9" target="_self" title="REWRITE1:def.9">Def9</a></i>;<br/>
<a NAME="E6:89"/><b>then </b>
<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> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a> 
 
<b>by </b><i><a class="ref" href="rewrite1.html#T45" target="_self" title="REWRITE1:th.45">Th45</a></i>;<br/>
<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="E8:89"/><i><font color="Green" title="E69">A5</font></i>: 
( <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <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="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <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/>
<b>take </b>
<font color="Maroon" title="c5">d</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/>

<b>thus </b><a NAME="E9:89"/>
( <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="txt" href="rewrite1.html#E8:89"><i><font color="Green" title="E69">A5</font></i></a>, <a class="ref" href="rewrite1.html#T22" target="_self" title="REWRITE1:th.22">Th22</a></i>;<br/>


</div>
