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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>, <font color="Olive" title="b4">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="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> 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> <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="Olive" title="b2">s</font>,<font color="Olive" title="b3">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> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b4">w</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>, <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> <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> 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> <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="Olive" title="b1">s</font>,<font color="Olive" title="b2">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> <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="c3">s</font>, <font color="Maroon" title="c4">t</font>, <font color="Maroon" title="c5">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="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> 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> <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> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c5">w</font> )</span><br/>









<b>assume </b><a NAME="E1:85"/><i><font color="Green" title="E55">A1</font></i>: 
<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>
 ; <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> <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> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c5">w</font></span><br/>

<a NAME="E2:85"/><i><font color="Green" title="E56">A2</font></i>: 
<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> <span class="p1">(<span class="default"><a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c5">w</font>
 
<b>by </b><i><a class="ref" href="rewrite2.html#T57" target="_self" title="REWRITE2:th.57">Th57</a>, <a class="ref" href="rewrite2.html#T60" target="_self" title="REWRITE2:th.60">Th60</a></i>;<br/>
<a NAME="E3:85"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</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="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:85"/>
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E1:85"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></i>;<br/>
<a NAME="E5:85"/><b>then </b>
<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> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:85"/><b>then </b>
<font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</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> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:85"/>
<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> <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> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c5">w</font>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:85"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="rewrite2.html#E3:85"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="rewrite2.html#T56" target="_self" title="REWRITE2:th.56">Th56</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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