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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="rewrite2.html#NM1" title="REWRITE2:NM.1">semi-Thue-system</a> of <font color="Maroon" title="c1">E</font><br/> for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</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>   st <font color="Olive" title="b2">s</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b1">S</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</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>   st <font color="Olive" title="b1">s</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b2">t</font>,<font color="Maroon" title="c2">S</font> holds <br/><font color="Olive" title="b1">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Olive" title="b2">t</font>,<font color="Maroon" title="c2">S</font></span><br/><b>let </b><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font> )</span><br/>







<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font></span><br/>

<b>take </b><font color="Maroon" title="c5">e</font> =  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">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>  st <br/>( <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="c5">e</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">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="c5">e</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b2">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b3">t1</font>,<font color="Maroon" title="c2">S</font> )</span><br/>

<i><font color="Green" title="E13">A2</font></i>: <font color="Maroon" title="c3">s</font> = 
<font color="Maroon" title="c3">s</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>
;<br/>
<font color="Maroon" title="c4">t</font> = 
<font color="Maroon" title="c4">t</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>
;<br/>
<b>hence </b><a NAME="E4:28"/>
 ex <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">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>  st <br/>( <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="c5">e</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">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="c5">e</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b2">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b3">t1</font>,<font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E2:28"><i><font color="Green" title="E13">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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