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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>, <font color="Olive" title="b2">T</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="b3">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#K3" title="CATALAN2:func.3">^omega</a>   st <font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b3">w</font> holds <br/><font color="Olive" title="b1">S</font>,<font color="Olive" title="b1">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <font color="Olive" title="b2">T</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b3">w</font></span><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>; <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">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#K3" title="CATALAN2:func.3">^omega</a>   st <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b1">w</font> holds <br/><font color="Maroon" title="c2">S</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> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b1">w</font></span><br/><b>let </b><font color="Maroon" title="c4">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#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="c2">S</font>,<font color="Maroon" title="c3">T</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c4">w</font> implies <font color="Maroon" title="c2">S</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> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c4">w</font> )</span><br/>







<b>assume </b><a NAME="E1:83"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c4">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</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> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c4">w</font></span><br/>

<b>thus </b><a NAME="E2:83"/>
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default"><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></span>)</span>
  <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<a NAME="E1:83_1"/><i><font color="Green" title="E56">A2</font></i>: 
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default"><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></span>)</span>
 
<b>by </b><i><a class="ref" href="rewrite2.html#T48" target="_self" title="REWRITE2:th.48">Th48</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E2:83_1"/>
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default"><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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default"><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></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font> )</span><br/>

<b>assume </b><a NAME="E1:83_1_1"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default"><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></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">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#K3" title="CATALAN2:func.3">^omega</a>  <b>by </b><i><a class="txt" href="#E1:83_1_1"><i><font color="Green" title="E57">A3</font></i></a></i>;<br/>
<a NAME="E3:83_1_1"/>
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c6">s</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="txt" href="#E1:83_1_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="rewrite2.html#T46" target="_self" title="REWRITE2:th.46">Th46</a></i>;<br/>
<a NAME="E4:83_1_1"/><b>then </b>
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E1:83"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="rewrite2.html#T59" target="_self" title="REWRITE2:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E5:83_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</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"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:83_1"/>
 <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite2.html#K10" title="REWRITE2:func.10">Lang</a> <font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default"><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></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:83_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


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