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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being   <a href="rewrite1.html#M2" title="REWRITE1:mode.2">Completion</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> iff  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">C</font> )</span><br/><b>let </b><font color="Maroon" title="c2">C</font> be    <a href="rewrite1.html#M2" title="REWRITE1:mode.2">Completion</a> of <font color="Maroon" title="c1">R</font>; <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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <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> iff  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b2">b</font>,<font color="Maroon" title="c2">C</font> )</span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</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"> ( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> iff  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">C</font> )</span><br/>





<a NAME="E1:95"/><i><font color="Green" title="E70">A1</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> iff <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c2">C</font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#D28" target="_self" title="REWRITE1:def.28">Def28</a></i>;<br/>
<a NAME="E2:95"/>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c2">C</font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#T38" target="_self" title="REWRITE1:th.38">Th38</a></i>;<br/>
<b>hence </b><a NAME="E3:95"/>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> implies  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D28" target="_self" title="REWRITE1:def.28">Def28</a>, <a class="ref" href="rewrite1.html#T56" target="_self" title="REWRITE1:th.56">Th56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<a NAME="E4:95"/>
(  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">C</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">C</font> &amp;  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">C</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">C</font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#T55" target="_self" title="REWRITE1:th.55">Th55</a></i>;<br/>
<a NAME="E5:95"/><b>then </b>
( <font color="Maroon" title="c2">C</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">a</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c2">C</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">b</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">C</font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:95"/>
(  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="#E1:95"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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