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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">w</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>  holds  <font color="Olive" title="b1">S</font>, <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Olive" title="b1">S</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b2">w</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">w</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>  holds  <font color="Maroon" title="c2">S</font>, <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b1">w</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">w</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="c2">S</font>, <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c3">w</font></span><br/>





<a NAME="E1:79"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_1"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span>
 ; <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">x</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E1:79_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a></span>;<br/>
<a NAME="E3:79_1"/>
<font color="Maroon" title="c3">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">s</font>, <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="rewrite2.html#T46" target="_self" title="REWRITE2:th.46">Th46</a></span>;<br/>
<a NAME="E4:79_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T43" target="_self" title="REWRITE2:th.43">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:79_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:79"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T22" target="_self" title="REWRITE2:th.22">Th22</a>, <a class="ref" href="rewrite2.html#T48" target="_self" title="REWRITE2:th.48">Th48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:79"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c3">w</font>,<span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite2.html#D9" target="_self" title="REWRITE2:def.9">REWRITE2:def 9</a> <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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