<?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>  st <font color="Olive" title="b1">S</font> is   <a href="rewrite2.html#NM2" title="REWRITE2:NM.2">Thue-system</a> of <font color="Maroon" title="c1">E</font> holds <br/> <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Olive" title="b1">S</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </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"> ( <font color="Maroon" title="c2">S</font> is   <a href="rewrite2.html#NM2" title="REWRITE2:NM.2">Thue-system</a> of <font color="Maroon" title="c1">E</font> implies  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">S</font> is   <a href="rewrite2.html#NM2" title="REWRITE2:NM.2">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">  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  ) &amp; ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:42_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <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="42_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:42_1_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  )
 <span class="kw">and </span><br/><a NAME="E4:42_1_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">b</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="#E3:42_1_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/>
<a NAME="E6:42_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E4:42_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="rewrite2.html#D6" target="_self" onmouseover="rs('rewrite2/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E7:42_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="rewrite2.html#T17" target="_self" onmouseover="rs('rewrite2/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E8:42_1_1"/><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="c7">b</font>,<font color="Maroon" title="c6">a</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>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D6" target="_self" onmouseover="rs('rewrite2/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:42_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</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><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:42_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_2"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:42_1_2"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  )
 <span class="kw">and </span><br/><a NAME="E4:42_1_2"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">b</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="#E3:42_1_2"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<a NAME="E6:42_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</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>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_2"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E4:42_1_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</a></span>;<br/>
<a NAME="E7:42_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D6" target="_self" onmouseover="rs('rewrite2/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E8:42_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="rewrite2.html#T17" target="_self" onmouseover="rs('rewrite2/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:42_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="rewrite2.html#D6" target="_self" onmouseover="rs('rewrite2/D6')" onmouseout="rh()">Def6</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:42"/>
 <a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K7" title="REWRITE2:func.7">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</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/>


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