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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">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#L3" title="ABCMIZ_0:struct.3">TAS-structure</a> ;<br/><b>let </b><font color="Maroon" title="c2">t1</font>, <font color="Maroon" title="c3">t2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice8.html#D1" title="LATTICE8:def.1">LATTICE8:def 1</a><br/>



<b>reconsider </b><font color="Maroon" title="c4">q1</font> = <font color="Maroon" title="c2">t1</font>, <font color="Maroon" title="c5">q2</font> = <font color="Maroon" title="c3">t2</font> as   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>assume </b><a NAME="E2:115"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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 consider </b><font color="Maroon" title="c6">a</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="E4:115"/>
not <font color="Maroon" title="c6">a</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="c5">q2</font>
 <b>and </b><br/><a NAME="E5:115"/><i><font color="Green" title="E70">A1</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c5">q2</font>
 <b>and </b><br/><a NAME="E6:115"/><i><font color="Green" title="E71">A2</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q1</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D31" target="_self" title="ABCMIZ_0:def.31">Def31</a></i>;<br/>
<a NAME="E7:115"/>
<font color="Maroon" title="c6">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c5">q2</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E5:115"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="abcmiz_0.html#D27" target="_self" title="ABCMIZ_0:def.27">Def27</a></i>;<br/>
<a NAME="E8:115"/><b>then </b>
<font color="Maroon" title="c4">q1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">q2</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E6:115"><i><font color="Green" title="E71">A2</font></i></a>, <a class="ref" href="abcmiz_0.html#T21" target="_self" title="ABCMIZ_0:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E9:115"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


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