<?xml version="1.0"?>
<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">u</font>, <font color="Olive" title="b5">v</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> &amp; <font color="Olive" title="b4">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Olive" title="b5">v</font>,<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> holds <br/><font color="Olive" title="b4">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b5">v</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">t</font>, <font color="Olive" title="b3">u</font>, <font color="Olive" title="b4">v</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> &amp; <font color="Olive" title="b3">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Olive" title="b4">v</font>,<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> holds <br/><font color="Olive" title="b3">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Olive" title="b4">v</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">t</font>, <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</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> &amp; <font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c6">v</font>,<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> implies <font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">S</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E37">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>
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c6">v</font>,<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 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="c5">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">S</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">u1</font>, <font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">s1</font>, <font color="Maroon" title="c10">t1</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> <span class="kw"> such that </span><br/><a NAME="E4:57"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">v1</font> )
 <span class="kw">and </span><br/><a NAME="E5:57"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<font color="Maroon" title="c9">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Maroon" title="c10">t1</font>,<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="#E2:57"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>;<br/>
<a NAME="E6:57"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">s1</font>,<font color="Maroon" title="c10">t1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="#E5:57"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>;<br/>
<a NAME="E7:57"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">s1</font>,<font color="Maroon" title="c10">t1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:57"/><span class="kw">then </span>
( <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:57"/>
<font color="Maroon" title="c5">u</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="rewrite2.html#T37" target="_self" onmouseover="rs('rewrite2/T37')" onmouseout="rh()">Th37</a></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"> verum</span><br/>


</div>
