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<b>let </b><font color="Maroon">T</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/><b>let </b><font color="Maroon">t1</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>, <font color="Maroon">t2</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>;<br/>





<b>set </b><font color="Maroon">R</font> = <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> ;<br/>
<a NAME="E1:132"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">T</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#T10" target="_self">Th10</a></i>;<br/>
<b>set </b><font color="Maroon">AA</font> = <span class="p1">{<span class="default"> <font color="Olive">t'</font> where <font color="Olive">t'</font> is   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> :  ex <font color="Olive">A</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Olive">t'</font> &amp; <font color="Olive">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:132"/>
( <font color="Maroon">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:132"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">A</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:132"/>
 <a href="rewrite1.html#K2">nf</a> <font color="Maroon">t2</font>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </span>)</span> <a href="rewrite1.html#R4">is_a_normal_form_of</a> <font color="Maroon">t2</font>,<font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> 
 
<b>by </b><i>, <a class="ref" href="rewrite1.html#T55">REWRITE1:55</a></i>;<br/>
<a NAME="E6:132"/><b>then </b>
<font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">t2</font>, <a href="rewrite1.html#K2">nf</a> <font color="Maroon">t2</font>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D6">REWRITE1:def 6</a></i>;<br/>
<a NAME="E7:132"/><b>then </b>
<font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">t2</font>, <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font>
 
;<br/>
<b>then consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E9:132"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">B</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon">v1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:132"/><i><font color="Green">E46</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">v1</font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="abcmiz_0.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">v2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:132"/><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">v2</font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T11" target="_self">Th11</a>, <a class="ref" href="abcmiz_0.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E14:132"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T13" target="_self">Th13</a>, , <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E15:132"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">v1</font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> &amp; <font color="Maroon">v2</font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T13" target="_self">Th13</a>, , </i>;<br/>
<a NAME="E16:132"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="abcmiz_0.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E17:132"/><b>then </b>
<font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T13" target="_self">Th13</a>, , <a class="ref" href="abcmiz_0.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E18:132"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> &amp; <font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T14" target="_self">Th14</a>, <a class="ref" href="abcmiz_0.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B</font></span>)</span> <a href="abcmiz_0.html#K6" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_0.html#T14" target="_self">Th14</a>, </i>
<br/>.= 
<font color="Maroon">v2</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_0.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<font color="Maroon">t2</font>
<b>by </b><i><a class="ref" href="abcmiz_0.html#T11" target="_self">Th11</a>, , <a class="ref" href="abcmiz_0.html#T16" target="_self">Th16</a>, </i>
;<br/>
<a NAME="E20:132"/><b>then </b>
( <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t'</font> where <font color="Olive">t'</font> is   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> :  ex <font color="Olive">A</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Olive">t'</font> &amp; <font color="Olive">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) </span>}</span>  &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">{<span class="default"> <font color="Olive">t'</font> where <font color="Olive">t'</font> is   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> :  ex <font color="Olive">A</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Olive">t'</font> &amp; <font color="Olive">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) </span>}</span> ,<font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D15" target="_self">Def15</a>, </i>;<br/>
<a NAME="E21:132"/><b>then </b>
<font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">t'</font> where <font color="Olive">t'</font> is   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> :  ex <font color="Olive">A</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Olive">t'</font> &amp; <font color="Olive">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) </span>}</span> ,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/>
<a NAME="E22:132"/><b>then </b>
( <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<b>hence </b><a NAME="E23:132"/>
<font color="Maroon">t1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div>
