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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> ;<br/>

<b>set </b><font color="Maroon" title="c2">R</font> = <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a> ;<br/>
<a NAME="E1:127"/>
<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#V9" title="REWRITE1:attr.9">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" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D24" title="REWRITE1:def.24">REWRITE1:def 24</a><br/>

<b>assume </b><a NAME="E1:127_1"/><i><font color="Green" title="E81">A1</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">t1</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">t2</font> = <font color="Maroon" title="c5">c</font> as   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">a2</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:127_1"/><i><font color="Green" title="E82">A2</font></i>: 
( not <font color="Maroon" title="c9">a2</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="c7">t1</font> &amp; <font color="Maroon" title="c9">a2</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c7">t1</font> &amp; <font color="Maroon" title="c9">a2</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c7">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:127_1"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="abcmiz_0.html#D31" target="_self" title="ABCMIZ_0:def.31">Def31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">a3</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:127_1"/><i><font color="Green" title="E83">A3</font></i>: 
( not <font color="Maroon" title="c10">a3</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="c8">t2</font> &amp; <font color="Maroon" title="c10">a3</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c10">a3</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c8">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:127_1"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="abcmiz_0.html#D31" target="_self" title="ABCMIZ_0:def.31">Def31</a></i>;<br/>
<a NAME="E7:127_1"/>
( <font color="Maroon" title="c9">a2</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c7">t1</font> &amp; <font color="Maroon" title="c10">a3</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c8">t2</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E4:127_1"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:127_1"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="abcmiz_0.html#D27" target="_self" title="ABCMIZ_0:def.27">Def27</a></i>;<br/>
<a NAME="E8:127_1"/><b>then </b><i><font color="Green" title="E84">A4</font></i>: 
( <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">t1</font> &amp; <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">t2</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E4:127_1"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:127_1"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="abcmiz_0.html#T21" target="_self" title="ABCMIZ_0:th.21">Th21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E10:127_1"/><i><font color="Green" title="E85">A5</font></i>: 
( <font color="Maroon" title="c11">A</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c7">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c11">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:127_1"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="abcmiz_0.html#D30" target="_self" title="ABCMIZ_0:def.30">Def30</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E12:127_1"/><i><font color="Green" title="E86">A6</font></i>: 
( <font color="Maroon" title="c12">B</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c7">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">t2</font> &amp; <font color="Maroon" title="c12">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t2</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:127_1"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:127_1"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="abcmiz_0.html#D30" target="_self" title="ABCMIZ_0:def.30">Def30</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">tt</font> = <font color="Maroon" title="c7">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">t2</font>;<br/>
<b>take </b>
<font color="Maroon" title="c7">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">t2</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D7" title="REWRITE1:def.7">REWRITE1:def 7</a><br/>

<b>thus </b><a NAME="E13:127_1"/>
( <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="c4">b</font>,<font color="Maroon" title="c7">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">t2</font> &amp; <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="c5">c</font>,<font color="Maroon" title="c7">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">t2</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E10:127_1"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="abcmiz_0.html#E12:127_1"><i><font color="Green" title="E86">A6</font></i></a>, <a class="ref" href="abcmiz_0.html#T73" target="_self" title="ABCMIZ_0:th.73">Th73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:127"/><b>then </b>
<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#V9" title="REWRITE1:attr.9">locally-confluent</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></i>;<br/>
<b>hence </b><a NAME="E3:127"/>
( <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#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a> &amp; <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#V4" title="REWRITE1:attr.4">with_UN_property</a> )
 ;<br/>


</div>
