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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="E1:73"/>
<font color="Maroon" title="c2">S</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E2:73"/><b>then </b><i><font color="Green" title="E48">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"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></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="E3:73"/>
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c3">w</font>,<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> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></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="73_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="E1:73_1"/><i><font color="Green" title="E49">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"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then 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>  ;<br/>
<a NAME="E3:73_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>,<font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E1:73_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="rewrite2.html#T46" target="_self" title="REWRITE2:th.46">Th46</a></i>;<br/>
<a NAME="E4:73_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#T41" target="_self" title="REWRITE2:th.41">Th41</a></i>;<br/>
<b>hence </b><a NAME="E5:73_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="E4:73"/>
 <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"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:73"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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