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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:89_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="rewrite1.html#V7" target="_self">confluent</a>
 ;<br/><b>thus </b><a NAME="E2:89_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  is <a href="rewrite1.html#V6" target="_self">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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D21" target="_self">REWRITE1:def 21</a><br/>

<b>assume </b><a NAME="E1:89_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  )
 ;<br/>

<a NAME="E2:89_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E3:89_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rewrite1.html#R6" target="_self">are_divergent_wrt</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:89_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rewrite1.html#E1:89_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="rewrite1.html#D22" target="_self">Def22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:89_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7" target="_self">Def7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D9" target="_self">REWRITE1:def 9</a><br/>

<b>thus </b><a NAME="E7:89_1_1"/>
( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> ) &amp; ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="rewrite1.html#E6:89_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="rewrite1.html#T21" target="_self">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">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  holds <br/><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="rewrite1.html#R7" target="_self">are_convergent&lt;=1_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D21" target="_self">REWRITE1:def 21</a><br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E3:89"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></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="E4:89"/>
( ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E3:89"><i><font color="Green">E68</font></i></a>, <a class="ref" href="rewrite1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E5:89"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="rewrite1.html#R7" target="_self">are_convergent&lt;=1_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a> 
 
<b>by </b><i><a class="txt" href="rewrite1.html#E2:89"><i><font color="Green">E67</font></i></a>, <a class="ref" href="rewrite1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E6:89"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a> 
 
<b>by </b><i><a class="ref" href="rewrite1.html#T45" target="_self">Th45</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:89"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K18">[*]</a>  <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D7" target="_self">Def7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></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="E9:89"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E8:89"><i><font color="Green">E69</font></i></a>, <a class="ref" href="rewrite1.html#T22" target="_self">Th22</a></i>;<br/>


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