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<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:67_1_1"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">a</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="rewrite1.html#D14" target="_self" title="REWRITE1:def.14">REWRITE1:def 14</a><br/>

<b>assume </b><a NAME="E2:67_1_1"/><i><font color="Green" title="E57">A2</font></i>: 
<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>
 ;<br/>

<b>then reconsider </b><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/>
<b>set </b><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"/><i><font color="Green" title="E58">A3</font></i>: 
<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"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</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><br/>

<b>assume </b><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> 
 ;<br/>

<a NAME="E2:67_1_1_1"/><b>then </b>
 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/>
<b>hence </b><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>
 ;<br/>


</div><b>end;</b></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> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E2:67_1_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="rewrite1.html#T13" target="_self" title="REWRITE1:th.13">Th13</a></i>;<br/>
<a NAME="E6:67_1_1"/><b>then </b><i><font color="Green" title="E59">A4</font></i>: 
( <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>  &amp; <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a> <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E1:67_1_1"><i><font color="Green" title="E56">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:67_1_1"/><i><font color="Green" title="E60">A5</font></i>: 
( <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>  &amp; ( 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> ) )
 <b>by </b><i><a class="txt" href="rewrite1.html#E4:67_1_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="rewrite1.html#D16" target="_self" title="REWRITE1:def.16">Def16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D11" target="_self" title="REWRITE1:def.11">REWRITE1:def 11</a><br/>

<a NAME="E9:67_1_1"/><i><font color="Green" title="E61">A6</font></i>: 
 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> )
 
<b>by </b><i><a class="txt" href="rewrite1.html#E8:67_1_1"><i><font color="Green" title="E60">A5</font></i></a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><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>
<div class="add"><b>given </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E1:67_1_1_2"/><i><font color="Green" title="E62">A7</font></i>: 
<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>
 ;<br/><a NAME="E2:67_1_1_2"/>
( <font color="Maroon" title="c5">x</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> &amp; <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E4:67_1_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="rewrite1.html#E6:67_1_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="rewrite1.html#E8:67_1_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="rewrite1.html#E1:67_1_1_2"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="rewrite1.html#T16" target="_self" title="REWRITE1:th.16">Th16</a>, <a class="ref" href="relat_2.html#D10" title="RELAT_2:def.10">RELAT_2:def 10</a></i>;<br/><a NAME="E3:67_1_1_2"/><b>then </b>
( 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> &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="Maroon" title="c6">b</font> &amp; <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> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E9:67_1_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="rewrite1.html#E1:67_1_1_2"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="rewrite1.html#T17" target="_self" title="REWRITE1:th.17">Th17</a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a>, <a class="ref" href="relat_2.html#D2" title="RELAT_2:def.2">RELAT_2:def 2</a></i>;<br/><a NAME="E4:67_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <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>  )
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:67_1_1_2"><i><font color="Green" title="E62">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rewrite1.html#E8:67_1_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="rewrite1.html#E1:67_1_1_2"><i><font color="Green" title="E62">A7</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E11:67_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>
 <b>by </b><i><a class="txt" href="rewrite1.html#E9:67_1_1"><i><font color="Green" title="E61">A6</font></i></a></i>;<br/>


</div>
