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

<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:76_6_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="rewrite1.html#R3" target="_self">is_a_normal_form_wrt</a> <font color="Maroon">R</font> &amp; <font color="Olive">b</font> <a href="rewrite1.html#R3" target="_self">is_a_normal_form_wrt</a> <font color="Maroon">R</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">R</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 <b>and </b><br/><a NAME="E2:76_6_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/><font color="Olive">a</font> <a href="rewrite1.html#R9" target="_self">has_a_normal_form_wrt</a> <font color="Maroon">R</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D14" target="_self">REWRITE1:def 14</a>,<a class="ref" href="rewrite1.html#D19" target="_self">REWRITE1:def 19</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D23" target="_self">REWRITE1:def 23</a><br/>

<b>assume </b><a NAME="E3:76_6_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> </span>)</span> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D4" target="_self">REWRITE1:def 4</a><br/>

<i><font color="Green">E124</font></i>:  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4">~</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T33">RELAT_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T38">RELAT_1:38</a></i>
<br/>.= 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_6_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_6_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_6_1_2_1"><b>suppose </b></a><a NAME="E1:76_6_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_6_1_2_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_6_1_2_2"><b>suppose </b></a><a NAME="E1:76_6_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:76_6_1_2_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T12" target="_self">Th12</a>, <a class="ref" href="rewrite1.html#T13" target="_self">Th13</a>, <a class="txt" href="rewrite1.html#E2"><i><font color="Green">Lemma22</font></i></a></i>;<br/><a NAME="E3:76_6_1_2_2"/><b>then </b><i><font color="Green">E125</font></i>: 
( <font color="Maroon">a</font> <a href="rewrite1.html#R9" target="_self">has_a_normal_form_wrt</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="rewrite1.html#R9" target="_self">has_a_normal_form_wrt</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D4" target="_self">Def4</a></i>;<br/><b>then consider </b><font color="Maroon">a'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:76_6_1_2_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">a'</font> <a href="rewrite1.html#R4" target="_self">is_a_normal_form_of</a> <font color="Maroon">a</font>,<font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">b'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:76_6_1_2_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">b'</font> <a href="rewrite1.html#R4" target="_self">is_a_normal_form_of</a> <font color="Maroon">b</font>,<font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T14" target="_self">Th14</a>, </i>;<br/><a NAME="E8:76_6_1_2_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">a'</font> <a href="rewrite1.html#R3" target="_self">is_a_normal_form_wrt</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b'</font> <a href="rewrite1.html#R3" target="_self">is_a_normal_form_wrt</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T15" target="_self">Th15</a>, <a class="ref" href="rewrite1.html#T16" target="_self">Th16</a>, </i>;<br/><a NAME="E9:76_6_1_2_2"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">R</font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">b</font>,<font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T15" target="_self">Th15</a>, <a class="ref" href="rewrite1.html#T16" target="_self">Th16</a>, </i>;<br/><a NAME="E10:76_6_1_2_2"/><b>then </b>
( <font color="Maroon">a'</font>,<font color="Maroon">a</font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T12" target="_self">Th12</a>, , </i>;<br/><a NAME="E11:76_6_1_2_2"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a'</font>,<font color="Maroon">b</font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E12:76_6_1_2_2"/><b>then </b>
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E13:76_6_1_2_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D3" target="_self">Def3</a>, <a class="ref" href="rewrite1.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E14:76_6_1_2_2"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
