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<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">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>  holds <br/> ( <font color="Olive" title="b2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Olive" title="b3">w</font>,<font color="Olive" title="b1">S</font>) iff <font color="Olive" title="b3">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b2">s</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" 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">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>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c2">S</font>) iff <font color="Olive" title="b2">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font>) iff <font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">S</font> )</span><br/>







<span class="kw">thus </span><a NAME="E1:60"/>
( <font color="Maroon" title="c3">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font>) implies <font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c3">s</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="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">S</font>) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1"/>
<font color="Maroon" title="c3">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c4">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"> <font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E2:60_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">t</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 <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c2">S</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:60_1"/>
<font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c3">s</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>

<span class="kw">thus </span><a NAME="E2:60"/>
( <font color="Maroon" title="c4">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">Lang</a> (<font color="Maroon" title="c4">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/>


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