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<b>let </b><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">T</font> be   <a href="rewrite2.html#NM1" title="REWRITE2:NM.1">semi-Thue-system</a> of <font color="Maroon" title="c1">E</font>;<br/><b>let </b><font color="Maroon" title="c4">w</font>, <font color="Maroon" title="c5">s</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:84"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c4">w</font>
 <b>and </b><br/><a NAME="E2:84"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font>
 ;<br/>

<a NAME="E3:84"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E2:84"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="rewrite2.html#D7" target="_self" title="REWRITE2:def.7">Def7</a></i>;<br/>
<a NAME="E4:84"/><b>then </b>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E1:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="rewrite2.html#T58" target="_self" title="REWRITE2:th.58">Th58</a></i>;<br/>
<b>hence </b><a NAME="E5:84"/>
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D7" target="_self" title="REWRITE2:def.7">Def7</a></i>;<br/>


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