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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a> <a href="abcmiz_0.html#V1" target="_self">Noetherian</a> non <a href="abcmiz_0.html#V4" target="_self">void</a> <a href="abcmiz_0.html#V9" target="_self">adj-structured</a> <a href="abcmiz_0.html#V14" target="_self">commutative</a>  <a href="abcmiz_0.html#L3" target="_self">TAS-structure</a> ;<br/>

<a NAME="E1:134"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  is  <a href="rewrite1.html#V3">strongly-normalizing</a> <a href="rewrite1.html#V4">with_UN_property</a> <a href="rewrite1.html#V8">with_Church-Rosser_property</a> <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T70" target="_self">Th70</a>, <a class="ref" href="abcmiz_0.html#T78" target="_self">Th78</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E2:134"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:134"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<a NAME="E4:134"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:134"><i><font color="Green">E86</font></i></a>, <a class="ref" href="abcmiz_0.html#D31" target="_self">Def31</a>, <a class="ref" href="abcmiz_0.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E5:134"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#T13">REWRITE1:13</a>, <a class="ref" href="rewrite1.html#T16">REWRITE1:16</a></i>;<br/>
<a NAME="E6:134"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> 
 
<b>by </b><i><a class="ref" href="rewrite1.html#T26">REWRITE1:26</a></i>;<br/>
<a NAME="E7:134"/><b>then </b>
 <a href="rewrite1.html#K2">nf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E1:134"><i><font color="Green">E85</font></i></a>, <a class="ref" href="rewrite1.html#T56">REWRITE1:56</a></i>;<br/>
<b>hence </b> <a href="abcmiz_0.html#K12" target="_self">radix</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </span>)</span>

<br/>.= 
 <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>2</sub></font>

;<br/><br/>


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