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<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="firebrick">::  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>  implies <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V5" title="REWRITE1:attr.5">with_NF_property</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:74_4_1"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V5" title="REWRITE1:attr.5">with_NF_property</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D20" target="_self" title="REWRITE1:def.20">REWRITE1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">a</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="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:74_4_1"/><span class="lab"><font color="Green" title="E64">A14</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="rewrite1.html#R3" title="REWRITE1:pred.3">is_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">a</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:74_4_1"/>
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">a</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c3">a</font>,<font color="Maroon" title="c2">b</font></span><br/>

<a NAME="E4:74_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">a</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="#E1:74_4_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="rewrite1.html#D23" target="_self" title="REWRITE1:def.23">Def23</a></span>;<br/>
<a NAME="E5:74_4_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:74_4_1"/>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_4_1"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="rewrite1.html#T33" target="_self" title="REWRITE1:th.33">Th34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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