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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>, <font color="Olive" title="b3">t</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>   st <font color="Olive" title="b2">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">{</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#K4" title="LANG1:func.4">}</a></span> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>   st <font color="Olive" title="b1">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">{</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#K4" title="LANG1:func.4">}</a></span> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Olive" title="b3">w</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font>, <font color="Maroon" title="c5">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">{</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#K4" title="LANG1:func.4">}</a></span> <a href="rewrite2.html#R4" title="REWRITE2:pred.4">are_equivalent_wrt</a> <font color="Maroon" title="c5">w</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:73"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c2">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">{</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#K4" title="LANG1:func.4">}</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="E2:73"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="lang1.html#K4" title="LANG1:func.4">{</a><span class="default"><span class="p3"><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#K4" title="LANG1:func.4">}</a></span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c2">S</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="lang1.html#K4" title="LANG1:func.4">{</a><span class="default"><span class="p3"><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#K4" title="LANG1:func.4">}</a></span></span>)</span>) or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c2">S</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="lang1.html#K4" title="LANG1:func.4">{</a><span class="default"><span class="p3"><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#K4" title="LANG1:func.4">}</a></span></span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c2">S</font>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">u</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a></span>;<br/>
<a NAME="E3:73_1"/>
<font color="Maroon" title="c5">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">{</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#K4" title="LANG1:func.4">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="rewrite2.html#T46" target="_self" onmouseover="rs('rewrite2/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E4:73_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="rewrite2.html#T45" target="_self" onmouseover="rs('rewrite2/T45')" onmouseout="rh()">Th45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:73_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c2">S</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:73"/>
 <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="lang1.html#K4" title="LANG1:func.4">{</a><span class="default"><span class="p3"><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#K4" title="LANG1:func.4">}</a></span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T48" target="_self" onmouseover="rs('rewrite2/T48')" onmouseout="rh()">Th48</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:73"/>
 <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c2">S</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c5">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="lang1.html#K4" title="LANG1:func.4">{</a><span class="default"><span class="p3"><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#K4" title="LANG1:func.4">}</a></span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite2.html#D9" target="_self" onmouseover="rs('rewrite2/D9')" onmouseout="rh()">REWRITE2:def 9</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
