<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">T</font> be  <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> <a href="abcmiz_0.html#V9" target="_self">adj-structured</a>  <a href="abcmiz_0.html#L2" target="_self">TA-structure</a> ;<br/><b>let </b><font color="Maroon">t</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>, <font color="Maroon">b</font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E3:56"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">a</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">t</font></span>)</span></span>)</span> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">t'</font> being   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:56"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font> &amp; <font color="Maroon">t'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:56"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">t'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t'</font> )
 
<b>by </b><i>, <a class="txt" href="abcmiz_0.html#E1:56"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>thus </b><a NAME="E7:56"/>
<font color="Maroon">b</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">t'</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D15" target="_self">ABCMIZ_0:def 15</a><br/>

<b>thus </b><a NAME="E1:56_1"/>
( <font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font> &amp; <font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">t</font> )
 <b>by </b><i>, <a class="txt" href="abcmiz_0.html#E1:56"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E8:56"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">t</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E9:56"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">t'</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D15" target="_self">ABCMIZ_0:def 15</a><br/>

<a NAME="E1:56_2"/>
<font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:56_2"/>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:56"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/>

<b>thus </b><a NAME="E3:56_2"/>
<font color="Maroon">t'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<a NAME="E10:56"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t'</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:56_3"/>
<font color="Maroon">t'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:56_3"/><b>then </b>
( <font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font> &amp; <font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:56_3"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t'</font> &amp; <font color="Maroon">t'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E4:56_3"/><b>then </b>
( <font color="Maroon">t'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t'</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E5:56_3"/><b>then </b>
( <font color="Maroon">t'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">t'</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:56_3"/>
<font color="Maroon">t'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:56"/>
<font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:56"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E12:56"/><b>then </b>
( <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="abcmiz_0.html#E2:56"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E13:56"/><b>then </b>
( <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:56"><i><font color="Green">E35</font></i></a>, , </i>;<br/>
<a NAME="E14:56"/><b>then </b>
( <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/>
<a NAME="E15:56"/><b>then </b>
<font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:56"/><b>then </b>
for <font color="Olive">t'</font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>  st <font color="Olive">t'</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> holds <br/><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">t'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E17:56"/>
<font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


</div>
