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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="rewrite1.html#M2" target="_self">Completion</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<a NAME="E1:99"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#D28" target="_self">Def28</a></i>;<br/>
<a NAME="E2:99"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rewrite1.html#R5" target="_self">are_convergent_wrt</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#T38" target="_self">Th38</a></i>;<br/>
<b>hence </b><a NAME="E3:99"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">c<sub>1</sub></font> implies  <a href="rewrite1.html#K2" target="_self">nf</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rewrite1.html#K2" target="_self">nf</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D28" target="_self">Def28</a>, <a class="ref" href="rewrite1.html#T56" target="_self">Th56</a></i>;<br/>

<a NAME="E4:99"/>
(  <a href="rewrite1.html#K2" target="_self">nf</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R4" target="_self">is_a_normal_form_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rewrite1.html#K2" target="_self">nf</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R4" target="_self">is_a_normal_form_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E5:99"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>3</sub></font>, <a href="rewrite1.html#K2" target="_self">nf</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1" target="_self">reduces</a> <font color="Maroon">c<sub>4</sub></font>, <a href="rewrite1.html#K2" target="_self">nf</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="rewrite1.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:99"/>
(  <a href="rewrite1.html#K2" target="_self">nf</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rewrite1.html#K2" target="_self">nf</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rewrite1.html#R2" target="_self">are_convertible_wrt</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:99"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rewrite1.html#D7" target="_self">Def7</a></i>;<br/>


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