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<b>consider </b><font color="Maroon" title="c1">P</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="abcmiz_0.html#V2" title="ABCMIZ_0:attr.2">Mizar-widening-like</a> non <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a> <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a> <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a> <a href="abcmiz_0.html#V7" title="ABCMIZ_0:attr.7">strict</a> <a href="abcmiz_0.html#V8" title="ABCMIZ_0:attr.8">consistent</a> <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a> <a href="abcmiz_0.html#V10" title="ABCMIZ_0:attr.10">adjs-typed</a>  <a href="abcmiz_0.html#L2" title="ABCMIZ_0:struct.2">TA-structure</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">T</font> =  <a href="abcmiz_0.html#G3" title="ABCMIZ_0:aggr.3">TAS-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U3" title="ABCMIZ_0:sel.3">adj-map</a> of <font color="Maroon" title="c1">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" title="ABCMIZ_0:func.9">sub</a> <font color="Maroon" title="c1">P</font></span>)</span> #);<br/>
<a NAME="E2:111_1_1"/>
( not  <a href="abcmiz_0.html#G3" title="ABCMIZ_0:aggr.3">TAS-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U3" title="ABCMIZ_0:sel.3">adj-map</a> of <font color="Maroon" title="c1">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" title="ABCMIZ_0:func.9">sub</a> <font color="Maroon" title="c1">P</font></span>)</span> #) is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="abcmiz_0.html#G3" title="ABCMIZ_0:aggr.3">TAS-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U3" title="ABCMIZ_0:sel.3">adj-map</a> of <font color="Maroon" title="c1">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" title="ABCMIZ_0:func.9">sub</a> <font color="Maroon" title="c1">P</font></span>)</span> #),the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="abcmiz_0.html#G3" title="ABCMIZ_0:aggr.3">TAS-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U3" title="ABCMIZ_0:sel.3">adj-map</a> of <font color="Maroon" title="c1">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" title="ABCMIZ_0:func.9">sub</a> <font color="Maroon" title="c1">P</font></span>)</span> #) #) )
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">T</font> =  <a href="abcmiz_0.html#G3" title="ABCMIZ_0:aggr.3">TAS-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U3" title="ABCMIZ_0:sel.3">adj-map</a> of <font color="Maroon" title="c1">P</font>,<span class="p1">(<span class="default"><a href="abcmiz_0.html#K9" title="ABCMIZ_0:func.9">sub</a> <font color="Maroon" title="c1">P</font></span>)</span> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> non <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a> <a href="abcmiz_0.html#V11" title="ABCMIZ_0:attr.11">strict</a>  <a href="abcmiz_0.html#L3" title="ABCMIZ_0:struct.3">TAS-structure</a>  <b>by </b><i><a class="ref" href="abcmiz_0.html#D4" target="_self" title="ABCMIZ_0:def.4">Def4</a>, <a class="ref" href="tex_2.html#T4" title="TEX_2:th.4">TEX_2:4</a>, <a class="ref" href="waybel_8.html#T12" title="WAYBEL_8:th.12">WAYBEL_8:12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V2" title="ABCMIZ_0:attr.2">Mizar-widening-like</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V8" title="ABCMIZ_0:attr.8">consistent</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V10" title="ABCMIZ_0:attr.10">adjs-typed</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V12" title="ABCMIZ_0:attr.12">non-absorbing</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V13" title="ABCMIZ_0:attr.13">subjected</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V14" title="ABCMIZ_0:attr.14">commutative</a> )</span><br/>

<b>thus </b><a NAME="E4:111_1_1"/>
<font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V2" title="ABCMIZ_0:attr.2">Mizar-widening-like</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V8" title="ABCMIZ_0:attr.8">consistent</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V10" title="ABCMIZ_0:attr.10">adjs-typed</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V12" title="ABCMIZ_0:attr.12">non-absorbing</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V13" title="ABCMIZ_0:attr.13">subjected</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V14" title="ABCMIZ_0:attr.14">commutative</a> )</span><br/>

<a NAME="E5:111_1_1"/>
 <a href="abcmiz_0.html#G1" title="ABCMIZ_0:aggr.1">AdjectiveStr</a>(# the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">P</font>,the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c1">P</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#G1" title="ABCMIZ_0:aggr.1">AdjectiveStr</a>(# the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c3">T</font>,the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c3">T</font> #)
 
;<br/>
<b>hence </b><a NAME="E6:111_1_1"/>
( <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T5" target="_self" title="ABCMIZ_0:th.5">Th5</a>, <a class="ref" href="abcmiz_0.html#T6" target="_self" title="ABCMIZ_0:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V8" title="ABCMIZ_0:attr.8">consistent</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V10" title="ABCMIZ_0:attr.10">adjs-typed</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V12" title="ABCMIZ_0:attr.12">non-absorbing</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V13" title="ABCMIZ_0:attr.13">subjected</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V14" title="ABCMIZ_0:attr.14">commutative</a> )</span><br/>

<b>thus </b><a NAME="E7:111_1_1"/>
( <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V8" title="ABCMIZ_0:attr.8">consistent</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V10" title="ABCMIZ_0:attr.10">adjs-typed</a> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T8" target="_self" title="ABCMIZ_0:th.8">Th8</a>, <a class="ref" href="abcmiz_0.html#T9" target="_self" title="ABCMIZ_0:th.9">Th9</a>, <a class="ref" href="abcmiz_0.html#T17" target="_self" title="ABCMIZ_0:th.17">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V12" title="ABCMIZ_0:attr.12">non-absorbing</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V13" title="ABCMIZ_0:attr.13">subjected</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V14" title="ABCMIZ_0:attr.14">commutative</a> )</span><br/>

<a NAME="E8:111_1_1"/><i><font color="Green" title="E66">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">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" title="ABCMIZ_0:def.26">ABCMIZ_0:def 26</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V13" title="ABCMIZ_0:attr.13">subjected</a> &amp; <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V14" title="ABCMIZ_0:attr.14">commutative</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c4">a</font></span><br/><b>reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">a</font> as   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">P</font> ;<br/><b>thus </b> <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_0.html#D22" target="_self" title="ABCMIZ_0:def.22">Def22</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_0.html#D6" target="_self" title="ABCMIZ_0:def.6">Def6</a></i>
<br/>.= 
 <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="abcmiz_0.html#D22" target="_self" title="ABCMIZ_0:def.22">Def22</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:111_1_1"/>
<font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V13" title="ABCMIZ_0:attr.13">subjected</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">T</font> is <a href="abcmiz_0.html#V14" title="ABCMIZ_0:attr.14">commutative</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c3">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D25" target="_self" title="ABCMIZ_0:def.25">ABCMIZ_0:def 25</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c4">a</font> &amp; (  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">a</font> as   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">P</font> ;<br/>
<a NAME="E2:111_1_1_2"/>
(  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c5">b</font> &amp;  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c5">b</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T11" target="_self" title="ABCMIZ_0:th.11">Th11</a></i>;<br/>
<a NAME="E3:111_1_1_2"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E8:111_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#T26" title="YELLOW_0:th.26">YELLOW_0:26</a></i>;<br/>
<a NAME="E4:111_1_1_2"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D22" target="_self" title="ABCMIZ_0:def.22">Def22</a></i>;<br/>
<b>hence </b><a NAME="E5:111_1_1_2"/>
( <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c4">a</font> &amp; (  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="abcmiz_0.html#K10" title="ABCMIZ_0:func.10">sub</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p3">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c4">t1</font>, <font color="Maroon" title="c5">t2</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c3">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D30" target="_self" title="ABCMIZ_0:def.30">ABCMIZ_0:def 30</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c4">t1</font> &amp; <font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c4">t1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">t2</font> holds <br/> ex <font color="Olive" title="b2">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="c3">T</font> st <br/>( <font color="Olive" title="b2">A</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">t2</font> &amp; <font color="Olive" title="b2">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="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t2</font> )</span><br/><b>let </b><font color="Maroon" title="c6">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c4">t1</font> &amp; <font color="Maroon" title="c6">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c4">t1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">t2</font> implies  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="c3">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="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">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="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t2</font> ) )</span><br/>



<b>assume </b><a NAME="E11:111_1_1"/>
( <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="c4">t1</font> &amp; <font color="Maroon" title="c6">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c4">t1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">t2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c3">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="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">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="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t2</font> )</span><br/>

<b>take </b><font color="Maroon" title="c7">A</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">t2</font> &amp; <font color="Maroon" title="c7">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="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t2</font> )</span><br/>

<b>thus </b><a NAME="E12:111_1_1"/>
<font color="Maroon" title="c7">A</font> <a href="abcmiz_0.html#R6" title="ABCMIZ_0:pred.6">is_properly_applicable_to</a> <font color="Maroon" title="c4">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">t2</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T64" target="_self" title="ABCMIZ_0:th.64">Th64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">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="c4">t1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t2</font></span><br/>

<b>thus </b><a NAME="E13:111_1_1"/>
<font color="Maroon" title="c7">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="c4">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t2</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
