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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>, <font color="Maroon" title="c6">v</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="E1:65"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c6">v</font>,<span class="p1"><a href="lang1.html#K8" title="LANG1:func.8">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="lang1.html#K8" title="LANG1:func.8">}</a></span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">u1</font>, <font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">s1</font>, <font color="Maroon" title="c10">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="E4:65"/><i><font color="Green" title="E45">A3</font></i>: 
( <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="c7">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c9">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Maroon" title="c10">t1</font>,<span class="p1"><a href="lang1.html#K8" title="LANG1:func.8">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="lang1.html#K8" title="LANG1:func.8">}</a></span> )
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:65"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="rewrite2.html#D5" target="_self" title="REWRITE2:def.5">Def5</a></i>;<br/>
<a NAME="E5:65"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">s1</font>,<font color="Maroon" title="c10">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> <span class="p1"><a href="lang1.html#K8" title="LANG1:func.8">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="lang1.html#K8" title="LANG1:func.8">}</a></span>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E4:65"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="rewrite2.html#D4" target="_self" title="REWRITE2:def.4">Def4</a></i>;<br/>
<a NAME="E6:65"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">s1</font>,<font color="Maroon" title="c10">t1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:65"/><b>then </b>
( <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:65"/>
<font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:65"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="rewrite2.html#E4:65"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="rewrite2.html#T37" target="_self" title="REWRITE2:th.37">Th37</a></i>;<br/>


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