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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a> non <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a> <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a> <a href="abcmiz_0.html#V14" title="ABCMIZ_0:attr.14">commutative</a>  <a href="abcmiz_0.html#L3" title="ABCMIZ_0:struct.3">TAS-structure</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">t</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">a</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Olive" title="b1">t</font> holds <br/> <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <font color="Olive" title="b1">t</font></span><br/>

<a NAME="E1:132"/><i><font color="Green" title="E83">A1</font></i>: 
<font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  is  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a> <a href="rewrite1.html#V4" title="REWRITE1:attr.4">with_UN_property</a> <a href="rewrite1.html#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a> <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T70" target="_self" title="ABCMIZ_0:th.70">Th70</a>, <a class="ref" href="abcmiz_0.html#T78" target="_self" title="ABCMIZ_0:th.78">Th78</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</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> being  <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> holds <br/> <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <font color="Maroon" title="c2">t</font></span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font>; <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> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> implies  <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <font color="Maroon" title="c2">t</font> )</span><br/>



<b>assume </b><a NAME="E2:132"/><i><font color="Green" title="E84">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <font color="Maroon" title="c2">t</font></span><br/>

<a NAME="E3:132"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c2">t</font> or  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c2">t</font> )
 
;<br/>
<a NAME="E4:132"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>,<font color="Maroon" title="c2">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:132"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="abcmiz_0.html#D31" target="_self" title="ABCMIZ_0:def.31">Def31</a>, <a class="ref" href="abcmiz_0.html#T25" target="_self" title="ABCMIZ_0:th.25">Th25</a></i>;<br/>
<a NAME="E5:132"/><b>then </b>
<font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a>, <a class="ref" href="rewrite1.html#T16" title="REWRITE1:th.16">REWRITE1:16</a></i>;<br/>
<a NAME="E6:132"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c2">t</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a> 
 
<b>by </b><i><a class="ref" href="rewrite1.html#T26" title="REWRITE1:th.26">REWRITE1:26</a></i>;<br/>
<b>hence </b><a NAME="E7:132"/>
 <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K12" title="ABCMIZ_0:func.12">radix</a> <font color="Maroon" title="c2">t</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:132"><i><font color="Green" title="E83">A1</font></i></a>, <a class="ref" href="rewrite1.html#T56" title="REWRITE1:th.56">REWRITE1:56</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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