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<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:88_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">R</font> is <a href="rewrite1.html#V7" target="_self">confluent</a>
 ;<br/><b>thus </b><a NAME="E2:88_1"/>
<font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#NR14" target="_self">has_weak-Church-Rosser_property</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_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> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D24" target="_self">REWRITE1:def 24</a><br/>

<b>assume </b><a NAME="E1:88_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<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> <a href="finseq_1.html#K18">[*]</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  )
 ;<br/>

<a NAME="E2:88_1_1"/><b>then </b>
( <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E3"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<a NAME="E3:88_1_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R6" target="_self">are_divergent_wrt</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:88_1_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</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#T9" target="_self">Th9</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:88_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">b</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">d</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D7" target="_self">REWRITE1:def 7</a><br/>

<b>thus </b><a NAME="E7:88_1_1"/>
( <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T14" target="_self">Th14</a>, <a class="txt" href="rewrite1.html#E4"><i><font color="Green">Lemma25</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:88"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  holds <br/><font color="Olive">b</font>,<font color="Olive">c</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D24" target="_self">REWRITE1:def 24</a><br/>

<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="E3:88"/><i><font color="Green">E32</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/>

<a NAME="E4:88"/>
( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) &amp; ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<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> <a href="finseq_1.html#K18">[*]</a>  or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) )
 
<b>by </b><i><a class="ref" href="rewrite1.html#T14" target="_self">Th14</a>, <a class="txt" href="rewrite1.html#E3"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<a NAME="E5:88"/><b>then </b>
( <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> <a href="finseq_1.html#K18">[*]</a>  or <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> or <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#T13" target="_self">Th13</a>, , </i>;<br/>
<a NAME="E6:88"/><b>then </b>
<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> <a href="finseq_1.html#K18">[*]</a> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:88"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">R</font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">d</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D7" target="_self">REWRITE1:def 7</a><br/>

<b>thus </b><a NAME="E9:88"/>
( <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T15" target="_self">Th15</a>, <a class="txt" href="rewrite1.html#E4"><i><font color="Green">Lemma25</font></i></a></i>;<br/>


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