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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>





<a NAME="E1:128"/><span class="lab"><font color="Green" title="E75">A1</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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/>
<span class="kw">assume </span><a NAME="E2:128"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:128"/><span class="kw">then </span>
( <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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E75">A1</font></span></a>, <a class="ref" href="abcmiz_0.html#D31" target="_self" onmouseover="rs('abcmiz_0/D31')" onmouseout="rh()">Def31</a>, <a class="ref" href="abcmiz_0.html#T24" target="_self" onmouseover="rs('abcmiz_0/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<a NAME="E4:128"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T12" onmouseover="rs('rewrite1/T12')" onmouseout="rh()">REWRITE1:12</a>, <a class="ref" href="rewrite1.html#T15" onmouseover="rs('rewrite1/T15')" onmouseout="rh()">REWRITE1:15</a></span>;<br/>
<a NAME="E5:128"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A2</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T25" onmouseover="rs('rewrite1/T25')" onmouseout="rh()">REWRITE1:25</a></span>;<br/>
<a NAME="E6:128"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#T69" target="_self" onmouseover="rs('abcmiz_0/T69')" onmouseout="rh()">Th69</a>, <a class="ref" href="abcmiz_0.html#T77" target="_self" onmouseover="rs('abcmiz_0/T77')" onmouseout="rh()">Th77</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:128"/>
 <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="ref" href="rewrite1.html#T55" onmouseover="rs('rewrite1/T55')" onmouseout="rh()">REWRITE1:55</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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