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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a>  implies <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V2" title="REWRITE1:attr.2">weakly-normalizing</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_1_1"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V2" title="REWRITE1:attr.2">weakly-normalizing</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D14" target="_self" title="REWRITE1:def.14">REWRITE1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> implies <font color="Maroon" title="c2">a</font> <a href="rewrite1.html#R9" title="REWRITE1:pred.9">has_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:67_1_1"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="rewrite1.html#R9" title="REWRITE1:pred.9">has_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">X</font> =  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Y</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> </span>}</span> ;<br/>
<a NAME="E4:67_1_1"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> </span>}</span>  or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E2:67_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> st <br/>( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:67_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:67_1_1"/>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T13" target="_self" title="REWRITE1:th.13">Th13</a></span>;<br/>
<a NAME="E6:67_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1"><span class="lab"><font color="Green" title="E56">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:67_1_1"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E9:67_1_1"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">x</font> </span>}</span>  &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E4:67_1_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="rewrite1.html#D16" target="_self" title="REWRITE1:def.16">Def16</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">x</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D11" target="_self" title="REWRITE1:def.11">REWRITE1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E10:67_1_1"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_1_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a></span>;<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D5" target="_self" title="REWRITE1:def.5">REWRITE1:def 5</a>,<a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">REWRITE1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font></span>
<div class="add"><a NAME="E1:67_1_1_2"/>
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R2" title="RELAT_2:pred.2">is_irreflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="relat_2.html#D10" title="RELAT_2:def.10">RELAT_2:def 10</a></span>;<br/><a NAME="E2:67_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A7</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E8:67_1_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="relat_2.html#D2" title="RELAT_2:def.2">RELAT_2:def 2</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E4:67_1_1_2"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:67_1_1_2"/>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1_1_2"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="rewrite1.html#T16" target="_self" title="REWRITE1:th.16">Th16</a></span>;<br/><a NAME="E6:67_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A9</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_1_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a></span>;<br/><a NAME="E7:67_1_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1_1_2"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></span>;<br/><a NAME="E8:67_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_1_1_2"><span class="lab"><font color="Green" title="E63">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:67_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_1_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E4:67_1_1_2"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="txt" href="#E2:67_1_1_2"><span class="lab"><font color="Green" title="E61">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E12:67_1_1"/>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_1_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
