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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> 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="c3">w</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/>





<a NAME="E1:82"/>
<font color="Maroon" title="c2">S</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="rewrite2.html#T22" target="_self" title="REWRITE2:th.22">Th22</a></i>;<br/>
<a NAME="E2:82"/><b>then </b><i><font color="Green" title="E52">A1</font></i>: 
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rewrite2.html#T48" target="_self" title="REWRITE2:th.48">Th48</a></i>;<br/>
<a NAME="E3:82"/>
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">x</font> as    <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>  <b>by </b><i><a class="txt" href="rewrite2.html#E1:82_1"><i><font color="Green" title="E53">A2</font></i></a></i>;<br/>
<a NAME="E3:82_1"/>
<font color="Maroon" title="c3">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">s</font>, <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E1:82_1"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="rewrite2.html#T46" target="_self" title="REWRITE2:th.46">Th46</a></i>;<br/>
<a NAME="E4:82_1"/><b>then </b>
<font color="Maroon" title="c3">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#T43" target="_self" title="REWRITE2:th.43">Th43</a></i>;<br/>
<b>hence </b><a NAME="E5:82_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:82"/>
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:82"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite2.html#D9" target="_self" title="REWRITE2:def.9">REWRITE2:def 9</a><br/>


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