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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">s</font>, <font color="Maroon" title="c4">t</font>, <font color="Maroon" title="c5">u</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>given </b><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font>, <font color="Maroon" title="c8">s1</font>, <font color="Maroon" title="c9">t1</font> being    <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> such that </b><a NAME="E2:31"/><i><font color="Green" title="E13">A</font></i>: 
( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">w</font> &amp; <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">w</font> &amp; <font color="Maroon" title="c8">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Maroon" title="c9">t1</font>,<font color="Maroon" title="c2">S</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite2.html#D5" target="_self" title="REWRITE2:def.5">REWRITE2:def 5</a><br/>

<i><font color="Green" title="E14">B1</font></i>: <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">s1</font></span>)</span></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">w</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E2:31"><i><font color="Green" title="E13">A</font></i></a>, <a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">w</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></i>
;<br/>
<font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">t1</font></span>)</span></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">w</font>
<b>by </b><i><a class="txt" href="rewrite2.html#E2:31"><i><font color="Green" title="E13">A</font></i></a>, <a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">w</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></i>
;<br/>
<b>hence </b><a NAME="E5:31"/>
<font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c5">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:31"><i><font color="Green" title="E13">A</font></i></a>, <a class="txt" href="rewrite2.html#E3:31"><i><font color="Green" title="E14">B1</font></i></a>, <a class="ref" href="rewrite2.html#D5" target="_self" title="REWRITE2:def.5">defDeriv1</a></i>;<br/>

<a NAME="E6:31"/><i><font color="Green" title="E15">B2</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E2:31"><i><font color="Green" title="E13">A</font></i></a>, <a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></i>;<br/>
<a NAME="E7:31"/>
<font color="Maroon" title="c4">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E2:31"><i><font color="Green" title="E13">A</font></i></a>, <a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></i>;<br/>
<b>hence </b><a NAME="E8:31"/>
<font color="Maroon" title="c3">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c4">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:31"><i><font color="Green" title="E13">A</font></i></a>, <a class="txt" href="rewrite2.html#E6:31"><i><font color="Green" title="E15">B2</font></i></a>, <a class="ref" href="rewrite2.html#D5" target="_self" title="REWRITE2:def.5">defDeriv1</a></i>;<br/>


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