<?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#V4" title="REWRITE1:attr.4">with_UN_property</a>  &amp; <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V2" title="REWRITE1:attr.2">weakly-normalizing</a>  implies <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:75_6_1"/><span class="lab"><font color="Green" title="E63">A16</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="rewrite1.html#R3" title="REWRITE1:pred.3">is_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b2">b</font> <a href="rewrite1.html#R3" title="REWRITE1:pred.3">is_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:75_6_1"/><span class="lab"><font color="Green" title="E64">A17</font></span>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> holds <br/><font color="Olive" title="b1">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 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="ref" href="rewrite1.html#D19" target="_self" title="REWRITE1:def.19">REWRITE1:def 19</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#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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#D23" target="_self" title="REWRITE1:def.23">REWRITE1:def 23</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>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> implies <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:75_6_1"/><span class="lab"><font color="Green" title="E65">A18</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rewrite1.html#D4" target="_self" title="REWRITE1:def.4">REWRITE1:def 4</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>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font></span><br/>

<span class="lab"><font color="Green" title="E66">A19</font></span>:  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T33" title="RELAT_1:th.33">RELAT_1:33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></span>
<br/>.= 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_6_1_2"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:75_6_1_2_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</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>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:75_6_1_2_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T39" target="_self" title="REWRITE1:th.39">Th39</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:75_6_1_2_2"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</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>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font></span><br/><div class="add"><a NAME="E2:75_6_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_6_1"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="txt" href="#E4:75_6_1"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="ref" href="rewrite1.html#T19" target="_self" title="REWRITE1:th.19">Th19</a></span>;<br/><a NAME="E3:75_6_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R9" title="REWRITE1:pred.9">has_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_6_1"><span class="lab"><font color="Green" title="E64">A17</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">b9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:75_6_1_2_2"/><span class="lab"><font color="Green" title="E68">A21</font></span>: 
<font color="Maroon" title="c4">b9</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D11" target="_self" title="REWRITE1:def.11">Def11</a></span>;<br/><a NAME="E6:75_6_1_2_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_6_1"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="txt" href="#E4:75_6_1"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="txt" href="#E1:75_6_1_2_2"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="ref" href="rewrite1.html#T19" target="_self" title="REWRITE1:th.19">Th19</a></span>;<br/><a NAME="E7:75_6_1_2_2"/><span class="kw">then </span>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_6_1"><span class="lab"><font color="Green" title="E64">A17</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:75_6_1_2_2"/><span class="lab"><font color="Green" title="E69">A22</font></span>: 
<font color="Maroon" title="c5">a9</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 class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D11" target="_self" title="REWRITE1:def.11">Def11</a></span>;<br/><a NAME="E10:75_6_1_2_2"/><span class="lab"><font color="Green" title="E70">A23</font></span>: 
<font color="Maroon" title="c5">a9</font> <a href="rewrite1.html#R3" title="REWRITE1:pred.3">is_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:75_6_1_2_2"><span class="lab"><font color="Green" title="E69">A22</font></span></a>, <a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></span>;<br/><a NAME="E11:75_6_1_2_2"/><span class="lab"><font color="Green" title="E71">A24</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_6_1"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="ref" href="rewrite1.html#D4" target="_self" title="REWRITE1:def.4">Def4</a></span>;<br/><a NAME="E12:75_6_1_2_2"/><span class="lab"><font color="Green" title="E72">A25</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="c5">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:75_6_1_2_2"><span class="lab"><font color="Green" title="E69">A22</font></span></a>, <a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></span>;<br/><a NAME="E13:75_6_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c2">a</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T26" target="_self" title="REWRITE1:th.26">Th26</a></span>;<br/><a NAME="E14:75_6_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A26</font></span>: 
<font color="Maroon" title="c5">a9</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:75_6_1_2_2"><span class="lab"><font color="Green" title="E71">A24</font></span></a>, <a class="ref" href="rewrite1.html#T31" target="_self" title="REWRITE1:th.31">Th31</a></span>;<br/><a NAME="E15:75_6_1_2_2"/><span class="lab"><font color="Green" title="E74">A27</font></span>: 
<font color="Maroon" title="c4">b9</font> <a href="rewrite1.html#R3" title="REWRITE1:pred.3">is_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_6_1_2_2"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></span>;<br/><a NAME="E16:75_6_1_2_2"/><span class="lab"><font color="Green" title="E75">A28</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_6_1_2_2"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></span>;<br/><a NAME="E17:75_6_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">b9</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T26" target="_self" title="REWRITE1:th.26">Th26</a></span>;<br/><a NAME="E18:75_6_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_6_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="txt" href="#E10:75_6_1_2_2"><span class="lab"><font color="Green" title="E70">A23</font></span></a>, <a class="txt" href="#E15:75_6_1_2_2"><span class="lab"><font color="Green" title="E74">A27</font></span></a>, <a class="txt" href="#E14:75_6_1_2_2"><span class="lab"><font color="Green" title="E73">A26</font></span></a>, <a class="ref" href="rewrite1.html#T31" target="_self" title="REWRITE1:th.31">Th31</a></span>;<br/><span class="kw">hence </span><a NAME="E19:75_6_1_2_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:75_6_1_2_2"><span class="lab"><font color="Green" title="E72">A25</font></span></a>, <a class="txt" href="#E16:75_6_1_2_2"><span class="lab"><font color="Green" title="E75">A28</font></span></a>, <a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</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></div><span class="kw">end;</span></div>

</div>
