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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>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="E1:127"/>
<font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  is <a href="rewrite1.html#V9">locally-confluent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D24">REWRITE1:def 24</a><br/>

<b>assume </b><a NAME="E1:127_1"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">a</font>, <font color="Maroon">t1</font> = <font color="Maroon">b</font>, <font color="Maroon">t2</font> = <font color="Maroon">c</font> as   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon">a2</font> being   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:127_1"/><i><font color="Green">E35</font></i>: 
( not <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">a2</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">a2</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D12" target="_self">Def12</a>, </i>;<br/>
<b>consider </b><font color="Maroon">a3</font> being   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:127_1"/><i><font color="Green">E43</font></i>: 
( not <font color="Maroon">a3</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">a3</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">t2</font> &amp; <font color="Maroon">a3</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D12" target="_self">Def12</a>, </i>;<br/>
<a NAME="E7:127_1"/>
( <font color="Maroon">a2</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">a3</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">t2</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:127_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t2</font> )
 
<b>by </b><i>, , </i>;<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="E10:127_1"/><i><font color="Green">E47</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> <font color="Maroon">t2</font> &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> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> )
 <b>by </b><i>, </i>;<br/>
<b>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="E12:127_1"/><i><font color="Green">E48</font></i>: 
( <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> <font color="Maroon">t2</font> &amp; <font color="Maroon">B</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> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">tt</font> = <font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">t2</font>;<br/>
<b>take </b>
<font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">t2</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D7">REWRITE1:def 7</a><br/>

<b>thus </b><a NAME="E13:127_1"/>
( <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">b</font>,<font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">t2</font> &amp; <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">c</font>,<font color="Maroon">t1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, <a class="ref" href="abcmiz_0.html#D13" target="_self">Def13</a>, <a class="ref" href="abcmiz_0.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:127"/><b>then </b>
<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#V9">locally-confluent</a> <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:127"/>
( <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  is <a href="rewrite1.html#V8">with_Church-Rosser_property</a> &amp; <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  is <a href="rewrite1.html#V4">with_UN_property</a> )
 ;<br/>


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