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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"> ( <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>  )</span><br/>

<span class="kw">set </span><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="E1:122"/>
<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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="122_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D24" onmouseover="rs('rewrite1/D24')" onmouseout="rh()">REWRITE1:def 24</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"> (  not <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>  or  not <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>  or <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:122_1"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:122_1"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
<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> 
 ; <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="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">T</font> <a href="abcmiz_0.html#K11" title="ABCMIZ_0:func.11">@--&gt;</a> </span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122_1"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="txt" href="#E2:122_1"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:122_1"/>
 not <font color="Maroon" title="c9">a2</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="c7">t1</font>
 <span class="kw">and </span><br/><a NAME="E6:122_1"/><span class="lab"><font color="Green" title="E74">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:122_1"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:122_1"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="abcmiz_0.html#D31" target="_self" onmouseover="rs('abcmiz_0/D31')" onmouseout="rh()">Def31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">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/>
<span class="kw">take </span>
<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 class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D7" onmouseover="rs('rewrite1/D7')" onmouseout="rh()">REWRITE1:def 7</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"> ( <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> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">a3</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E9:122_1"/>
 not <font color="Maroon" title="c11">a3</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="c8">t2</font>
 <span class="kw">and </span><br/><a NAME="E10:122_1"/><span class="lab"><font color="Green" title="E76">A5</font></span>: 
<font color="Maroon" title="c11">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>
 <span class="kw">and </span><br/><a NAME="E11:122_1"/><span class="lab"><font color="Green" title="E77">A6</font></span>: 
<font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:122_1"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="abcmiz_0.html#D31" target="_self" onmouseover="rs('abcmiz_0/D31')" onmouseout="rh()">Def31</a></span>;<br/>
<a NAME="E12:122_1"/>
<font color="Maroon" title="c11">a3</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c8">t2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:122_1"><span class="lab"><font color="Green" title="E76">A5</font></span></a></span>;<br/>
<a NAME="E13:122_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:122_1"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="abcmiz_0.html#T20" target="_self" onmouseover="rs('abcmiz_0/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E14:122_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A7</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="Olive" title="b1">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:122_1"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="txt" href="#E7:122_1"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="abcmiz_0.html#D30" target="_self" onmouseover="rs('abcmiz_0/D30')" onmouseout="rh()">Def30</a></span>;<br/>
<a NAME="E15:122_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:122_1"><span class="lab"><font color="Green" title="E74">A3</font></span></a></span>;<br/>
<a NAME="E16:122_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:122_1"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="abcmiz_0.html#T20" target="_self" onmouseover="rs('abcmiz_0/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E17:122_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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="Olive" title="b1">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:122_1"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="txt" href="#E11:122_1"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="abcmiz_0.html#D30" target="_self" onmouseover="rs('abcmiz_0/D30')" onmouseout="rh()">Def30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:122_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:122_1"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="ref" href="abcmiz_0.html#T72" target="_self" onmouseover="rs('abcmiz_0/T72')" onmouseout="rh()">Th72</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/>


</div><span class="kw">end;</span></div>
<a NAME="E2:122"/><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>  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>
 
<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></span>;<br/>
<span class="kw">hence </span><a NAME="E3:122"/>
( <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>  )
 ; <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/>


</div>
