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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">s</font>, <font color="Maroon" title="c5">t</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><a NAME="E1:40"/>
<font color="Maroon" title="c4">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c5">t</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/>

<b>then consider </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><br/><a NAME="E3:40"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c4">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="c5">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> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Maroon" title="c3">T</font> )
 <b>by </b><i><a class="ref" href="rewrite2.html#D5" target="_self" title="REWRITE2:def.5">Def5</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E23">A2</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">t1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E3:40"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="rewrite2.html#D4" target="_self" title="REWRITE2:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">t1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">t1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> )
 </span><b>by </b><i><a class="txt" href="rewrite2.html#E4:40"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">t1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><a NAME="E2:40_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="rewrite2.html#D4" target="_self" title="REWRITE2:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:40_1_1"/>
( <font color="Maroon" title="c4">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c4">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c3">T</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E3:40"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="rewrite2.html#D5" target="_self" title="REWRITE2:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">t1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 ;<br/><div class="add"><a NAME="E2:40_1_2"/><b>then </b>
<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="c3">T</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D4" target="_self" title="REWRITE2:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:40_1_2"/>
( <font color="Maroon" title="c4">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c4">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c3">T</font> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E3:40"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="rewrite2.html#D5" target="_self" title="REWRITE2:def.5">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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