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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>   st <font color="Olive" title="b2">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">S</font> holds <br/> ex <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font>, <font color="Olive" title="b5">s1</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#K2" title="CATALAN2:func.2">^omega</a>  st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b5">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b4">w</font> &amp; <font color="Olive" title="b5">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b5">s1</font>,<font color="Olive" title="b1">S</font> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</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#K2" title="CATALAN2:func.2">^omega</a>   st <font color="Olive" title="b1">s</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c2">S</font> holds <br/> ex <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font>, <font color="Olive" title="b4">s1</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#K2" title="CATALAN2:func.2">^omega</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b4">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">w</font> &amp; <font color="Olive" title="b4">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b4">s1</font>,<font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</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#K2" title="CATALAN2:func.2">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c3">s</font>,<font color="Maroon" title="c2">S</font> implies  ex <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font>, <font color="Olive" title="b3">s1</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#K2" title="CATALAN2:func.2">^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="Olive" title="b1">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">w</font> &amp; <font color="Olive" title="b3">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b3">s1</font>,<font color="Maroon" title="c2">S</font> ) )</span><br/>





<span class="kw">given </span><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font>, <font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c7">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#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><a NAME="E2:28"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<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="c4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font>
 <span class="kw">and </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<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="c4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font>
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c6">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Maroon" title="c7">t1</font>,<font color="Maroon" title="c2">S</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font>, <font color="Olive" title="b3">s1</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#K2" title="CATALAN2:func.2">^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="Olive" title="b1">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">w</font> &amp; <font color="Olive" title="b3">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b3">s1</font>,<font color="Maroon" title="c2">S</font> )</span><br/>

<a NAME="E5:28"/>
<font color="Maroon" title="c4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="afinsq_1.html#T31" title="AFINSQ_1:th.31">AFINSQ_1:31</a></span>;<br/>
<a NAME="E6:28"/><span class="kw">then </span>
<font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T31" title="AFINSQ_1:th.31">AFINSQ_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:28"/>
 ex <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font>, <font color="Olive" title="b3">s1</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#K2" title="CATALAN2:func.2">^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="Olive" title="b1">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">w</font> &amp; <font color="Olive" title="b3">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Olive" title="b3">s1</font>,<font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E14">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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