<?xml version="1.0"?>
<div class="add">

<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> holds   <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></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">  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span><br/>



<a NAME="E1:49"/><i><font color="Green" title="E29">A1</font></i>: 
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span> or <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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green" title="E31">A3</font></i>: 
( <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="catalan2.html#K3" title="CATALAN2:func.3">^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="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  &amp; <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> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><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="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>  <b>by </b><i><a class="txt" href="rewrite2.html#E3:49_1"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<a NAME="E5:49_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> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E1:49_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="rewrite2.html#E3:49_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">w</font>, <font color="Maroon" title="c10">s1</font>, <font color="Maroon" title="c11">t1</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> <b> such that </b><br/><a NAME="E7:49_1"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">w</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c11">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">w</font> &amp; <font color="Maroon" title="c10">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Maroon" title="c11">t1</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#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="rewrite2.html#D5" target="_self" title="REWRITE2:def.5">Def5</a></i>;<br/>
<a NAME="E8:49_1"/><i><font color="Green" title="E33">A5</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c11">t1</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> <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#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rewrite2.html#E7:49_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="rewrite2.html#D4" target="_self" title="REWRITE2:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c11">t1</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> <font color="Maroon" title="c2">S</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c11">t1</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#K8" title="REWRITE2:func.8">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span> )
 </span><b>by </b><i><a class="txt" href="rewrite2.html#E8:49_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c11">t1</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> <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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:49_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">s1</font> <a href="rewrite2.html#R1" title="REWRITE2:pred.1">--&gt;.</a> <font color="Maroon" title="c11">t1</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#D4" target="_self" title="REWRITE2:def.4">Def4</a></i>;<br/><a NAME="E3:49_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">s1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">w</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c11">t1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="rewrite2.html#T15" target="_self" title="REWRITE2:th.15">Th15</a></i>;<br/><a NAME="E4:49_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <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#E3:49_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="rewrite2.html#E7:49_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E5:49_1_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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c11">t1</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#K8" title="REWRITE2:func.8">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:49_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/><a NAME="E3:49_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E3:49_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="rewrite2.html#E7:49_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/><b>hence </b><a NAME="E4:49_1_1_2"/>
<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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:49"/>
<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="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:49_2"/><i><font color="Green" title="E30">A6</font></i>: 
<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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1"/>
( <font color="Maroon" title="c3">x</font> <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> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span> )
 </span><b>by </b><i><a class="txt" href="rewrite2.html#E1:49_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:49_2_1_1"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <font color="Maroon" title="c2">S</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite2.html#T26" target="_self" title="REWRITE2:th.26">Th26</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E3:49_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:49_2_1_1"><i><font color="Green" title="E31">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:49_2_1_2"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite2.html#T27" target="_self" title="REWRITE2:th.27">Th27</a></i>;<br/><a NAME="E3:49_2_1_2"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite2.html#T26" target="_self" title="REWRITE2:th.26">Th26</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:49_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E2:49_2_1_2"><i><font color="Green" title="E31">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">==&gt;.-relation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="rewrite2.html#K8" title="REWRITE2:func.8">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rewrite2.html#E1:49"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
