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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a></span><br/>

<a NAME="E1:91"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> st <br/>( <font color="Olive" title="b2">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b3">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Olive" title="b3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>; <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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1"/>
( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">X</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> ) )</span><br/><div class="add"><a NAME="E2:91_1_1_1"/><b>then </b>
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/> <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">q</font>
 ;<br/><a NAME="E3:91_1_1_1"/><b>then </b><i><font color="Green" title="E51">A1</font></i>: 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/><a NAME="E4:91_1_1_1"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span></span><br/>

<a NAME="E2:91_1_1_1_1"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> st <br/>for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c4">CO</font> =  <a href="conlat_1.html#K5" title="CONLAT_1:func.5">Concept-with-all-Objects</a> <font color="Maroon" title="c1">C</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#T35" target="_self" title="CONLAT_1:th.35">Th35</a></i>;<br/>
<a NAME="E2:91_1_1_1_1_1"/>
for <font color="Olive" title="b1">CP</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> holds <br/> ( <font color="Maroon" title="c4">CO</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CO</font> &amp; <font color="Olive" title="b1">CP</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">CO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CO</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">CP</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">CO</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CO</font> &amp; <font color="Maroon" title="c5">CP</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">CO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CO</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">CP</font> = <font color="Maroon" title="c5">CP</font> as  <a href="conlat_1.html#V6" title="CONLAT_1:attr.6">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="conlat_1.html#T35" target="_self" title="CONLAT_1:th.35">Th35</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">CO</font> = <font color="Maroon" title="c4">CO</font> as  <a href="conlat_1.html#V6" title="CONLAT_1:attr.6">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" title="CONLAT_1:func.10">B-join</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">CO</font>,<font color="Maroon" title="c6">CP</font> = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" title="CONLAT_1:func.10">B-join</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">CP</font>,<font color="Maroon" title="c7">CO</font>
<b>by </b><i><a class="ref" href="conlat_1.html#T37" target="_self" title="CONLAT_1:th.37">Th37</a></i>
<br/>.= 
<font color="Maroon" title="c7">CO</font>
<b>by </b><i><a class="ref" href="conlat_1.html#T43" target="_self" title="CONLAT_1:th.43">Th43</a></i>
;<br/>
<b>hence </b><a NAME="E4:91_1_1_1_1_1_1"/>
( <font color="Maroon" title="c4">CO</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CO</font> &amp; <font color="Maroon" title="c5">CP</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">CO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CO</font> )
 <b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</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>hence </b><a NAME="E3:91_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> st <br/>for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> )
 ; <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>
<a NAME="E3:91_1_1_1_1"/><b>then </b>
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a></i>;<br/>
<a NAME="E4:91_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#D17" title="LATTICES:def.17">LATTICES:def 17</a></i>;<br/>
<b>hence </b><a NAME="E5:91_1_1_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</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>hence </b><a NAME="E5:91_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> ) )
 <b>by </b><i><a class="txt" href="conlat_1.html#E3:91_1_1_1"><i><font color="Green" title="E51">A1</font></i></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_2"/>
<font color="Maroon" title="c2">X</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> 
 ; <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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> ) )</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c2">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> ;<br/><b>set </b><font color="Maroon" title="c4">ExX</font> = <span class="p1">{<span class="default"> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> ;<br/><a NAME="E3:91_1_1_2"/><i><font color="Green" title="E51">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> </span><br/>

<a NAME="E1:91_1_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E3:91_1_1_2_1"/>
the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:91_1_1_2_1"/>
the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> 
 ; <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>then reconsider </b><font color="Maroon" title="c5">ExX</font> = <span class="p1">{<span class="default"> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>set </b><font color="Maroon" title="c6">E1</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font>;<br/><a NAME="E5:91_1_1_2"/><i><font color="Green" title="E52">A3</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_2"/><i><font color="Green" title="E53">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">Y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">ExX</font>;<br/>
<a NAME="E3:91_1_1_2_2"/>
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ExX</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c9">Y'</font> being    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E5:91_1_1_2_2"/><i><font color="Green" title="E54">A5</font></i>: 
( <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c9">Y'</font> &amp; <font color="Maroon" title="c9">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ;<br/>
<a NAME="E6:91_1_1_2_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c9">Y'</font>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_2"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="conlat_1.html#E5:91_1_1_2_2"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:91_1_1_2_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 ; <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><a NAME="E6:91_1_1_2"/>
for <font color="Olive" title="b1">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font> iff for <font color="Olive" title="b2">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b2">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">o</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</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">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font> iff for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> )</span><br/>

<a NAME="E1:91_1_1_2_3"/><i><font color="Green" title="E53">A6</font></i>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font> implies for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1_1_2_3_1"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font>
 ; <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">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font></span><br/>

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</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">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c8">x</font></span><br/>

<a NAME="E2:91_1_1_2_3_1"/>
the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ExX</font>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E3:91_1_1_2"><i><font color="Green" title="E51">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:91_1_1_2_3_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_3_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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>
<a NAME="E2:91_1_1_2_3"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1_1_2_3_2"/><i><font color="Green" title="E54">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</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">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font></span><br/>

<a NAME="E2:91_1_1_2_3_2"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ExX</font> holds <br/><font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ExX</font> implies <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_3_2_1"/>
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ExX</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">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font></span><br/>

<a NAME="E2:91_1_1_2_3_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">Y'</font> being   <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> st <br/>( <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">Y'</font> &amp; <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 
;<br/>
<b>hence </b><a NAME="E3:91_1_1_2_3_2_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_3_2"><i><font color="Green" title="E54">A8</font></i></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>hence </b><a NAME="E3:91_1_1_2_3_2"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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>hence </b><a NAME="E3:91_1_1_2_3"/>
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c5">ExX</font> iff for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_3"><i><font color="Green" title="E53">A6</font></i></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>then consider </b><font color="Maroon" title="c7">O</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E8:91_1_1_2"/><i><font color="Green" title="E53">A9</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> iff for <font color="Olive" title="b2">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b2">x</font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E5:91_1_1_2"><i><font color="Green" title="E52">A3</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c8">InX</font> = <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c9">In</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> ;<br/><a NAME="E9:91_1_1_2"/><i><font color="Green" title="E54">A10</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font></span><br/>

<b>then consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:91_1_1_2_4"/><i><font color="Green" title="E55">A11</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">Y'</font> being    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E5:91_1_1_2_4"/><i><font color="Green" title="E56">A12</font></i>: 
( <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c12">Y'</font> &amp; <font color="Maroon" title="c12">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E3:91_1_1_2_4"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:91_1_1_2_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E3:91_1_1_2_4"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="conlat_1.html#E5:91_1_1_2_4"><i><font color="Green" title="E56">A12</font></i></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><a NAME="E10:91_1_1_2"/>
for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  iff  ex <font color="Olive" title="b2">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b2">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  iff  ex <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> )</span><br/>

<a NAME="E1:91_1_1_2_5"/><i><font color="Green" title="E55">A13</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  implies  ex <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1_1_2_5_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> 
 ; <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">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:91_1_1_2_5_1"/><i><font color="Green" title="E56">A14</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">Y'</font> being    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E5:91_1_1_2_5_1"/><i><font color="Green" title="E57">A15</font></i>: 
( <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c12">Y'</font> &amp; <font color="Maroon" title="c12">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E3:91_1_1_2_5_1"><i><font color="Green" title="E56">A14</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:91_1_1_2_5_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E3:91_1_1_2_5_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="conlat_1.html#E5:91_1_1_2_5_1"><i><font color="Green" title="E57">A15</font></i></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>
<a NAME="E2:91_1_1_2_5"/>
(  ex <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> implies <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1_1_2_5_2"/><i><font color="Green" title="E56">A16</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> </span><br/>

<a NAME="E2:91_1_1_2_5_2"/>
 ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_5_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c11">x</font> being    <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font><b> such that </b><br/><a NAME="E2:91_1_1_2_5_2_1"/><i><font color="Green" title="E57">A17</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_5_2"><i><font color="Green" title="E56">A16</font></i></a></i>;<br/>
<a NAME="E3:91_1_1_2_5_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
;<br/>
<a NAME="E4:91_1_1_2_5_2_1"/><b>then </b>
the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:91_1_1_2_5_2_1"/>
 ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="conlat_1.html#E2:91_1_1_2_5_2_1"><i><font color="Green" title="E57">A17</font></i></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>hence </b><a NAME="E3:91_1_1_2_5_2"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>hence </b><a NAME="E3:91_1_1_2_5"/>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="conlat_1.html#M2" title="CONLAT_1:mode.2">Element</a> of  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  iff  ex <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_5"><i><font color="Green" title="E55">A13</font></i></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>then consider </b><font color="Maroon" title="c10">A'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E12:91_1_1_2"/><i><font color="Green" title="E55">A18</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b2">x</font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E9:91_1_1_2"><i><font color="Green" title="E54">A10</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E14:91_1_1_2"/><i><font color="Green" title="E56">A19</font></i>: 
<font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font></span>)</span>
 ;<br/><b>set </b><font color="Maroon" title="c12">p</font> =  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c11">A</font> #);<br/><a NAME="E15:91_1_1_2"/>
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c11">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:91_1_1_2_6"/>
<font color="Maroon" title="c7">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1"><b>proof </b></a><div class="add">

<a NAME="E1:91_1_1_2_6_1"/><i><font color="Green" title="E57">A20</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> iff for <font color="Olive" title="b2">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b2">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">o</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> iff for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> )</span><br/>

<a NAME="E1:91_1_1_2_6_1_1"/><i><font color="Green" title="E58">A21</font></i>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> implies for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1_1_2_6_1_1_1"/><i><font color="Green" title="E59">A22</font></i>: 
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font>
 ; <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">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font></span><br/>

<a NAME="E2:91_1_1_2_6_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c14">x</font></span><br/>

<a NAME="E1:91_1_1_2_6_1_1_1_1"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E8:91_1_1_2"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="conlat_1.html#E1:91_1_1_2_6_1_1_1"><i><font color="Green" title="E59">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:91_1_1_2_6_1_1_1_1"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#D13" target="_self" title="CONLAT_1:def.13">Def13</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>hence </b><a NAME="E3:91_1_1_2_6_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font>
 ; <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>
<a NAME="E2:91_1_1_2_6_1_1"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1_1_2_6_1_1_2"/><i><font color="Green" title="E59">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font></span><br/>

<a NAME="E2:91_1_1_2_6_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c14">x</font></span><br/>

<a NAME="E1:91_1_1_2_6_1_1_2_1"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_6_1_1_2"><i><font color="Green" title="E59">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:91_1_1_2_6_1_1_2_1"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#D13" target="_self" title="CONLAT_1:def.13">Def13</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>hence </b><a NAME="E3:91_1_1_2_6_1_1_2"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E8:91_1_1_2"><i><font color="Green" title="E53">A9</font></i></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>hence </b><a NAME="E3:91_1_1_2_6_1_1"/>
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> iff for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_6_1_1"><i><font color="Green" title="E58">A21</font></i></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>
<a NAME="E2:91_1_1_2_6_1"/><i><font color="Green" title="E58">A24</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> implies <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_6_1_2"/><i><font color="Green" title="E59">A25</font></i>: 
<font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c13">x'</font> as   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E3:91_1_1_2_6_1_2"/>
for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> holds <br/><font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">a</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> implies <font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c15">a</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_6_1_2_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c15">a</font></span><br/>

<b>then consider </b><font color="Maroon" title="c16">x</font> being    <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font><b> such that </b><br/><a NAME="E3:91_1_1_2_6_1_2_1"/><i><font color="Green" title="E60">A26</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E12:91_1_1_2"><i><font color="Green" title="E55">A18</font></i></a></i>;<br/>
<a NAME="E4:91_1_1_2_6_1_2_1"/>
<font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_6_1"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="conlat_1.html#E1:91_1_1_2_6_1_2"><i><font color="Green" title="E59">A25</font></i></a></i>;<br/>
<a NAME="E5:91_1_1_2_6_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">o</font> where <font color="Olive" title="b1">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b2">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c16">x</font> holds <br/><font color="Olive" title="b1">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D7" target="_self" title="CONLAT_1:def.7">Def7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">y</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E7:91_1_1_2_6_1_2_1"/><i><font color="Green" title="E61">A27</font></i>: 
( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c16">x</font> holds <br/><font color="Maroon" title="c17">y</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font> ) )
 ;<br/>
<b>thus </b><a NAME="E8:91_1_1_2_6_1_2_1"/>
<font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E3:91_1_1_2_6_1_2_1"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="conlat_1.html#E7:91_1_1_2_6_1_2_1"><i><font color="Green" title="E61">A27</font></i></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>
<a NAME="E4:91_1_1_2_6_1_2"/><b>then </b>
<font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">o</font> where <font color="Olive" title="b1">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b2">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> holds <br/><font color="Olive" title="b1">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:91_1_1_2_6_1_2"/>
<font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#D7" target="_self" title="CONLAT_1:def.7">Def7</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>
<a NAME="E3:91_1_1_2_6_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font> implies <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_6_1_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font></span><br/>

<a NAME="E2:91_1_1_2_6_1_3"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">o</font> where <font color="Olive" title="b1">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b2">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> holds <br/><font color="Olive" title="b1">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D7" target="_self" title="CONLAT_1:def.7">Def7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">x'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:91_1_1_2_6_1_3"/><i><font color="Green" title="E59">A28</font></i>: 
( <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font> holds <br/><font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font> ) )
 ;<br/>
<a NAME="E5:91_1_1_2_6_1_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c15">x</font></span><br/>

<a NAME="E1:91_1_1_2_6_1_3_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c15">x</font> holds <br/><font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_6_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">a</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c15">x</font> implies <font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c16">a</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_6_1_3_1_1"/>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c15">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c16">a</font></span><br/>

<a NAME="E2:91_1_1_2_6_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A'</font>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E12:91_1_1_2"><i><font color="Green" title="E55">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:91_1_1_2_6_1_3_1_1"/>
<font color="Maroon" title="c14">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c16">a</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E4:91_1_1_2_6_1_3"><i><font color="Green" title="E59">A28</font></i></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>
<a NAME="E2:91_1_1_2_6_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">o</font> where <font color="Olive" title="b1">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b2">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c15">x</font> holds <br/><font color="Olive" title="b1">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:91_1_1_2_6_1_3_1"/>
<font color="Maroon" title="c14">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#D7" target="_self" title="CONLAT_1:def.7">Def7</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>hence </b><a NAME="E6:91_1_1_2_6_1_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_6_1"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="conlat_1.html#E4:91_1_1_2_6_1_3"><i><font color="Green" title="E59">A28</font></i></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>hence </b><a NAME="E4:91_1_1_2_6_1"/>
<font color="Maroon" title="c7">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A'</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E2:91_1_1_2_6_1"><i><font color="Green" title="E58">A24</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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>hence </b><a NAME="E2:91_1_1_2_6"/>
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c11">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E14:91_1_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="ref" href="conlat_1.html#T22" target="_self" title="CONLAT_1:th.22">Th22</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>then reconsider </b><font color="Maroon" title="c13">p</font> =  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c11">A</font> #) as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#T35" target="_self" title="CONLAT_1:th.35">Th35</a></i>;<br/><a NAME="E17:91_1_1_2"/><i><font color="Green" title="E57">A29</font></i>: 
<font color="Maroon" title="c13">p</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_7"><b>proof </b></a><div class="add">

<a NAME="E1:91_1_1_2_7"/>
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c13">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c13">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">q</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_7_1"/><i><font color="Green" title="E58">A30</font></i>: 
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">q</font></span><br/>

<a NAME="E2:91_1_1_2_7_1"/>
the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_7_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span></span><br/>

<b>hence </b><a NAME="E2:91_1_1_2_7_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">q</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <b>by </b><i><a class="txt" href="conlat_1.html#E8:91_1_1_2"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="conlat_1.html#E1:91_1_1_2_7_1"><i><font color="Green" title="E58">A30</font></i></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>
<a NAME="E3:91_1_1_2_7_1"/><b>then </b>
<font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="conlat_1.html#R2" title="CONLAT_1:pred.2">is-SubConcept-of</a> <font color="Maroon" title="c14">q</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D19" target="_self" title="CONLAT_1:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E4:91_1_1_2_7_1"/>
<font color="Maroon" title="c13">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">q</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#T47" target="_self" title="CONLAT_1:th.47">Th47</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>hence </b><a NAME="E2:91_1_1_2_7"/>
<font color="Maroon" title="c13">p</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</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><a NAME="E18:91_1_1_2"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c14">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">p</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_8"/><i><font color="Green" title="E58">A31</font></i>: 
<font color="Maroon" title="c14">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">p</font></span><br/>

<a NAME="E2:91_1_1_2_8"/>
the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> or <font color="Maroon" title="c15">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_2_8_1"/><i><font color="Green" title="E59">A32</font></i>: 
<font color="Maroon" title="c15">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c15">x'</font> as   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E3:91_1_1_2_8_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="conlat_1.html#M3" title="CONLAT_1:mode.3">Element</a> of <font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c17">x</font></span><br/>

<a NAME="E1:91_1_1_2_8_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<a NAME="E3:91_1_1_2_8_1_1"/>
<font color="Maroon" title="c14">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c18">x</font>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_8"><i><font color="Green" title="E58">A31</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E4:91_1_1_2_8_1_1"/><b>then </b>
<font color="Maroon" title="c14">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="conlat_1.html#R2" title="CONLAT_1:pred.2">is-SubConcept-of</a> <font color="Maroon" title="c18">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#T47" target="_self" title="CONLAT_1:th.47">Th47</a></i>;<br/>
<a NAME="E5:91_1_1_2_8_1_1"/><b>then </b><i><font color="Green" title="E60">A33</font></i>: 
the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D19" target="_self" title="CONLAT_1:def.19">Def19</a></i>;<br/>
<b>thus </b><a NAME="E6:91_1_1_2_8_1_1"/>
<font color="Maroon" title="c16">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:91_1_1_2_8_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="txt" href="conlat_1.html#E5:91_1_1_2_8_1_1"><i><font color="Green" title="E60">A33</font></i></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>hence </b><a NAME="E4:91_1_1_2_8_1"/>
<font color="Maroon" title="c15">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 <b>by </b><i><a class="txt" href="conlat_1.html#E8:91_1_1_2"><i><font color="Green" title="E53">A9</font></i></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>
<a NAME="E3:91_1_1_2_8"/><b>then </b>
<font color="Maroon" title="c14">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="conlat_1.html#R2" title="CONLAT_1:pred.2">is-SubConcept-of</a> <font color="Maroon" title="c13">p</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D19" target="_self" title="CONLAT_1:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E4:91_1_1_2_8"/>
<font color="Maroon" title="c14">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">p</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#T47" target="_self" title="CONLAT_1:th.47">Th47</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>hence </b><a NAME="E19:91_1_1_2"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> ) )
 <b>by </b><i><a class="txt" href="conlat_1.html#E17:91_1_1_2"><i><font color="Green" title="E57">A29</font></i></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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:91"/>
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D6" title="VECTSP_8:def.6">VECTSP_8:def 6</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>
