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

<b>consider </b><font color="Maroon">P</font> being  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V2">reflexive</a> <a href="lattice3.html#V3">complete</a> <a href="abcmiz_0.html#V2" target="_self">Mizar-widening-like</a> non <a href="abcmiz_0.html#V4" target="_self">void</a> <a href="abcmiz_0.html#V5" target="_self">involutive</a> <a href="abcmiz_0.html#V6" target="_self">without_fixpoints</a> <a href="abcmiz_0.html#V7" target="_self">strict</a> <a href="abcmiz_0.html#V8" target="_self">consistent</a> <a href="abcmiz_0.html#V9" target="_self">adj-structured</a> <a href="abcmiz_0.html#V10" target="_self">adjs-typed</a>  <a href="abcmiz_0.html#L2" target="_self">TA-structure</a> ;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="abcmiz_0.html#G3" target="_self">TAS-structure</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">P</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" target="_self">sub</a> <font color="Maroon">P</font></span>)</span> #);<br/>
<a NAME="E2:112_1_1"/>
( not  <a href="abcmiz_0.html#G3" target="_self">TAS-structure</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">P</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" target="_self">sub</a> <font color="Maroon">P</font></span>)</span> #) is <a href="struct_0.html#V3">empty</a> &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="abcmiz_0.html#G3" target="_self">TAS-structure</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">P</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" target="_self">sub</a> <font color="Maroon">P</font></span>)</span> #),the <a href="orders_2.html#U1">InternalRel</a> of <a href="abcmiz_0.html#G3" target="_self">TAS-structure</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">P</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" target="_self">sub</a> <font color="Maroon">P</font></span>)</span> #) #) )
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">T</font> =  <a href="abcmiz_0.html#G3" target="_self">TAS-structure</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">P</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U3" target="_self">adj-map</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" target="_self">sub</a> <font color="Maroon">P</font></span>)</span> #) as  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V2">reflexive</a> non <a href="abcmiz_0.html#V4" target="_self">void</a> <a href="abcmiz_0.html#V11" target="_self">strict</a>  <a href="abcmiz_0.html#L3" target="_self">TAS-structure</a>  <b>by </b><i>, <a class="ref" href="tex_2.html#T4">TEX_2:4</a>, <a class="ref" href="waybel_8.html#T12">WAYBEL_8:12</a></i>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E4:112_1_1"/>
<font color="Maroon">T</font> is <a href="abcmiz_0.html#V2" target="_self">Mizar-widening-like</a>
 ;<br/>

<a NAME="E5:112_1_1"/>
 <a href="abcmiz_0.html#G1" target="_self">AdjectiveStr</a>(# the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">P</font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">P</font> #) <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#G1" target="_self">AdjectiveStr</a>(# the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font>,the <a href="abcmiz_0.html#U2" target="_self">non-op</a> of <font color="Maroon">T</font> #)
 
;<br/>
<b>hence </b><a NAME="E6:112_1_1"/>
( <font color="Maroon">T</font> is <a href="abcmiz_0.html#V5" target="_self">involutive</a> &amp; <font color="Maroon">T</font> is <a href="abcmiz_0.html#V6" target="_self">without_fixpoints</a> )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E7:112_1_1"/>
( <font color="Maroon">T</font> is <a href="abcmiz_0.html#V8" target="_self">consistent</a> &amp; <font color="Maroon">T</font> is <a href="abcmiz_0.html#V9" target="_self">adj-structured</a> &amp; <font color="Maroon">T</font> is <a href="abcmiz_0.html#V10" target="_self">adjs-typed</a> )
 <b>by </b><i>, , </i>;<br/>

<a NAME="E8:112_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #)
 
;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D26" target="_self">ABCMIZ_0:def 26</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">P</font> ;<br/><b>thus </b> <a href="abcmiz_0.html#K10" target="_self">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span> = 
 <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> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <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> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="abcmiz_0.html#K10" target="_self">sub</a> <font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:112_1_1"/>
<font color="Maroon">T</font> is <a href="abcmiz_0.html#V13" target="_self">subjected</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D25" target="_self">ABCMIZ_0:def 25</a><br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">P</font> ;<br/>
<a NAME="E2:112_1_1_2"/>
(  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">b</font> &amp;  <a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:112_1_1_2"/><b>then </b>
( <span class="p1">(<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#K1">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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#K1">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<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#K1">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E4:112_1_1_2"/><b>then </b>
 <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#K1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <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#K1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<a NAME="E5:112_1_1_2"/><b>then </b>
 <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#K1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#K10" target="_self">sub</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:112_1_1_2"/>
( <span class="p1">(<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#K1">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="abcmiz_0.html#K10" target="_self">sub</a> <font color="Maroon">a</font> &amp; (  <a href="abcmiz_0.html#K3" target="_self">types</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="abcmiz_0.html#K10" target="_self">sub</a> <font color="Maroon">a</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#K1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" target="_self">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" target="_self">non-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>


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

<b>let </b><font color="Maroon">t1</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>, <font color="Maroon">t2</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="abcmiz_0.html#D30" target="_self">ABCMIZ_0:def 30</a><br/><b>let </b><font color="Maroon">a</font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E11:112_1_1"/>
( <font color="Maroon">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">t2</font> )
 ;<br/>

<b>take </b><font color="Maroon">A</font> =  <a href="subset_1.html#K1">{}</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font>;<br/>

<b>thus </b><a NAME="E12:112_1_1"/>
<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>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T6" target="_self">Th6</a></i>;<br/>

<b>thus </b><a NAME="E13:112_1_1"/>
<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">t2</font>
 <b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


</div>
