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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">C</font> be   <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1"/>
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c3">f</font> being    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V3" title="LATTICE4:attr.3">isomorphism</a>
 <b>by </b><i><a class="ref" href="lattice4.html#D5" title="LATTICE4:def.5">LATTICE4:def 5</a></i>;<br/><a NAME="E4:22_1"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V1" title="LATTICE4:attr.1">monomorphism</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="lattice4.html#D4" title="LATTICE4:def.4">LATTICE4:def 4</a></i>;<br/><b>take </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span>;<br/><b>take </b><font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span>;<br/><b>thus </b><a NAME="E5:22_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g</font> is <a href="lattice6.html#V8" title="LATTICE6:attr.8">supremum-dense</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice6.html#D13" title="LATTICE6:def.13">LATTICE6:def 13</a><br/>

<b>consider </b><font color="Maroon" title="c7">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="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E2:22_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="lattice4.html#T9" title="LATTICE4:th.9">LATTICE4:9</a></i>;<br/>
<a NAME="E3:22_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> is <a href="lattice6.html#V8" title="LATTICE6:attr.8">supremum-dense</a>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T12" target="_self" title="CONLAT_2:th.12">Th12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">D'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:22_1_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c8">D'</font>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice6.html#D13" title="LATTICE6:def.13">LATTICE6:def 13</a></i>;<br/>
<a NAME="E6:22_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c8">D'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c7">b</font> &amp; ( for <font color="Olive" title="b1">r</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="c2">C</font></span>)</span>  st <font color="Maroon" title="c8">D'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c7">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font> ) )
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> ;<br/>
<a NAME="E7:22_1_1"/><i><font color="Green" title="E20">A6</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><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><br/>

<b>assume </b><a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x'</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="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x'</font> &amp; <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:22_1_1_1"/><i><font color="Green" title="E22">A8</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:22_1_1_1"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="conlat_2.html#E5:22_1_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E7:22_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:22_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_1_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E9:22_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E6:22_1_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_1_1"/><i><font color="Green" title="E21">A10</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_1_1_2"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">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="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1_2"/><i><font color="Green" title="E22">A11</font></i>: 
( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q'</font> &amp; <font color="Maroon" title="c11">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> )
 ;<br/>
<a NAME="E4:22_1_1_2"/>
<font color="Maroon" title="c11">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E6:22_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_1_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1_1_2"/>
<font color="Maroon" title="c10">q</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conlat_2.html#E2:22_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_1_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:22_1_1"/>
for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b2">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c6">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:22_1_1_3"/><i><font color="Green" title="E22">A12</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c10">r</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">r'</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="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_1_3"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">r'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="lattice4.html#T9" title="LATTICE4:th.9">LATTICE4:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">r'</font> = <font color="Maroon" title="c11">r'</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="c2">C</font></span>)</span> ;<br/>
<a NAME="E5:22_1_1_3"/>
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="c2">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="c8">D'</font> holds <br/><font color="Olive" title="b1">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">r'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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="c2">C</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_1_1_3_1"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font>
 ;<br/>

<a NAME="E2:22_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> 
 
;<br/>
<a NAME="E3:22_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">r'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E1:22_1_1_3"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_1_1_3"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_1_3_1"/>
<font color="Maroon" title="c13">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">r'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">D'</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c12">r'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E7:22_1_1_3"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">r'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E8:22_1_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conlat_2.html#E2:22_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_1_1_3"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:22_1_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E8:22_1_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E11:22_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g</font></span>)</span> st <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E7:22_1_1"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:22_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">d</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice6.html#D14" title="LATTICE6:def.14">LATTICE6:def 14</a><br/>

<b>consider </b><font color="Maroon" title="c7">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="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E2:22_1_2"/><i><font color="Green" title="E17">A14</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="lattice4.html#T9" title="LATTICE4:th.9">LATTICE4:9</a></i>;<br/>
<a NAME="E3:22_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T12" target="_self" title="CONLAT_2:th.12">Th12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">D'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:22_1_2"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c8">D'</font>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice6.html#D14" title="LATTICE6:def.14">LATTICE6:def 14</a></i>;<br/>
<a NAME="E6:22_1_2"/><i><font color="Green" title="E19">A16</font></i>: 
( <font color="Maroon" title="c7">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c8">D'</font> &amp; ( for <font color="Olive" title="b1">r</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="c2">C</font></span>)</span>  st <font color="Olive" title="b1">r</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c8">D'</font> holds <br/><font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b</font> ) )
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_1_2"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> ;<br/>
<a NAME="E7:22_1_2"/><i><font color="Green" title="E20">A17</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><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><br/>

<b>assume </b><a NAME="E1:22_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x'</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="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_2_1"/><i><font color="Green" title="E21">A18</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x'</font> &amp; <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:22_1_2_1"/><i><font color="Green" title="E22">A19</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x'</font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1_2_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:22_1_2_1"/><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_1_2_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="conlat_2.html#E5:22_1_2_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E7:22_1_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:22_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_1_2_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E9:22_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E6:22_1_2_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_1_2"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_1_2_2"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">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="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_2_2"/><i><font color="Green" title="E22">A22</font></i>: 
( <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q'</font> &amp; <font color="Maroon" title="c11">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> )
 ;<br/>
<a NAME="E4:22_1_2_2"/>
<font color="Maroon" title="c7">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">q'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E6:22_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_1_2_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E5:22_1_2_2"/>
<font color="Maroon" title="c6">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c10">q</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conlat_2.html#E2:22_1_2"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_1_2_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:22_1_2"/>
for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">r</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b2">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span>  holds <br/><font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:22_1_2_3"/><i><font color="Green" title="E22">A23</font></i>: 
<font color="Maroon" title="c10">r</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">r'</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="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E3:22_1_2_3"/><i><font color="Green" title="E23">A24</font></i>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">r'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="lattice4.html#T9" title="LATTICE4:th.9">LATTICE4:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">r'</font> = <font color="Maroon" title="c11">r'</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="c2">C</font></span>)</span> ;<br/>
<a NAME="E5:22_1_2_3"/>
<font color="Maroon" title="c12">r'</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c8">D'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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="c2">C</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_1_2_3_1"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font>
 ;<br/>

<a NAME="E2:22_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> 
 
;<br/>
<a NAME="E3:22_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">r'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E1:22_1_2_3"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_1_2_3"><i><font color="Green" title="E23">A24</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_2_3_1"/>
<font color="Maroon" title="c12">r'</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c13">q</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_1_2_3"/><b>then </b>
<font color="Maroon" title="c12">r'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_1_2"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E7:22_1_2_3"/>
<font color="Maroon" title="c10">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conlat_2.html#E2:22_1_2"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_1_2_3"><i><font color="Green" title="E23">A24</font></i></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:22_1_2"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <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="c2">C</font></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E8:22_1_2"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E11:22_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">d</font></span>)</span> st <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E7:22_1_2"><i><font color="Green" title="E20">A17</font></i></a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c6">o</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>;<br/><b>let </b><font color="Maroon" title="c7">a</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>;<br/><a NAME="E7:22_1"/><i><font color="Green" title="E17">A25</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c7">a</font> iff <span class="p1">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="conlat_2.html#T13" target="_self" title="CONLAT_2:th.13">Th13</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_3"/><i><font color="Green" title="E18">A26</font></i>: 
<font color="Maroon" title="c6">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c7">a</font>
 ;<br/><a NAME="E2:22_1_3"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:22_1_3"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <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_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o</font> &amp; <font color="Maroon" title="c3">f</font> <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_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E4:22_1_3"/>
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conlat_2.html#E7:22_1"><i><font color="Green" title="E17">A25</font></i></a>, <a class="txt" href="conlat_2.html#E1:22_1_3"><i><font color="Green" title="E18">A26</font></i></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E9:22_1"/><i><font color="Green" title="E18">A27</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>
 ;<br/><a NAME="E10:22_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:22_1"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <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_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o</font> &amp; <font color="Maroon" title="c3">f</font> <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_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E12:22_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_2.html#K4" title="CONLAT_2:func.4">gamma</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" title="CONLAT_2:func.5">delta</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="conlat_2.html#E9:22_1"><i><font color="Green" title="E18">A27</font></i></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</a></i>;<br/><b>hence </b><a NAME="E13:22_1"/>
<font color="Maroon" title="c6">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T13" target="_self" title="CONLAT_2:th.13">Th13</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c3">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c4">d</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c2">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E2:22"/><i><font color="Green" title="E15">A28</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g</font> is <a href="lattice6.html#V8" title="LATTICE6:attr.8">supremum-dense</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">d</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a> &amp; ( 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="c2">C</font><br/> 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="c2">C</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> iff <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> ) ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<font color="Olive" title="b3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">O</font> is   <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="c2">C</font>, <font color="Olive" title="b2">A</font> is   <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="c2">C</font>, <font color="Olive" title="b3">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">o</font></span>)</span> where <font color="Olive" title="b4">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> ) </span>}</span> ;<br/>
<a NAME="E3:22"/><i><font color="Green" title="E16">A29</font></i>: 
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c2">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10" title="CONLAT_1:func.10">B-join</a> <font color="Maroon" title="c2">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9" title="CONLAT_1:func.9">B-meet</a> <font color="Maroon" title="c2">C</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="conlat_1.html#D23" title="CONLAT_1:def.23">CONLAT_1:def 23</a></i>;<br/>
<a NAME="E4:22"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<font color="Olive" title="b3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">O</font> is   <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="c2">C</font>, <font color="Olive" title="b2">A</font> is   <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="c2">C</font>, <font color="Olive" title="b3">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">o</font></span>)</span> where <font color="Olive" title="b4">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</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><br/>

<b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<font color="Olive" title="b3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">O</font> is   <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="c2">C</font>, <font color="Olive" title="b2">A</font> is   <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="c2">C</font>, <font color="Olive" title="b3">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">o</font></span>)</span> where <font color="Olive" title="b4">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> ) </span>}</span> 
 ;<br/>

<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="c2">C</font>, <font color="Maroon" title="c8">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="c2">C</font>, <font color="Maroon" title="c9">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:22_2"/><i><font color="Green" title="E17">A30</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c8">A</font> #),<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c8">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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> </span>}</span> ,<font color="Maroon" title="c1">L</font> )
 ;<br/>
<a NAME="E4:22_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="c8">A</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22"><i><font color="Green" title="E16">A29</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_2"><i><font color="Green" title="E17">A30</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E5:22_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_2"><i><font color="Green" title="E17">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<font color="Olive" title="b3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">O</font> is   <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="c2">C</font>, <font color="Olive" title="b2">A</font> is   <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="c2">C</font>, <font color="Olive" title="b3">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">o</font></span>)</span> where <font color="Olive" title="b4">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> ) </span>}</span>  as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E6:22"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</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><br/>

<b>assume </b><a NAME="E1:22_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span>
 ;<br/>

<a NAME="E2:22_3"/><b>then </b><i><font color="Green" title="E17">A31</font></i>: 
<font color="Maroon" title="c7">y</font> is  <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="c2">C</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22"><i><font color="Green" title="E16">A29</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">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="c2">C</font>, <font color="Maroon" title="c9">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="c2">C</font><b> such that </b><br/><a NAME="E4:22_3"/><i><font color="Green" title="E18">A32</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c8">O'</font>,<font color="Maroon" title="c9">A'</font> #)
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">z</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">O'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E6:22_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_3"><i><font color="Green" title="E17">A31</font></i></a>, <a class="txt" href="conlat_2.html#E4:22_3"><i><font color="Green" title="E18">A32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:22_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22"/><b>then </b><i><font color="Green" title="E17">A33</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:22"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_4"/><i><font color="Green" title="E18">A34</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">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="c2">C</font>, <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="c2">C</font>, <font color="Maroon" title="c12">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:22_4"/><i><font color="Green" title="E19">A35</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<font color="Maroon" title="c12">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c10">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="c2">C</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">O</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c13">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="c2">C</font>, <font color="Maroon" title="c14">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="c2">C</font>, <font color="Maroon" title="c15">z'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:22_4"/><i><font color="Green" title="E20">A36</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<font color="Maroon" title="c15">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c15">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">O'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E1:22_4"><i><font color="Green" title="E18">A34</font></i></a></i>;<br/>
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_4"><i><font color="Green" title="E19">A35</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #)
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_4"><i><font color="Green" title="E20">A36</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c8">y1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_4"><i><font color="Green" title="E19">A35</font></i></a>, <a class="txt" href="conlat_2.html#E5:22_4"><i><font color="Green" title="E20">A36</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c9">y2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="conlat_2.html#E7:22"><i><font color="Green" title="E17">A33</font></i></a>, <a class="ref" href="funct_1.html#D1" title="FUNCT_1:def.1">FUNCT_1:def 1</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:22"/><i><font color="Green" title="E18">A37</font></i>: 
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c2">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10" title="CONLAT_1:func.10">B-join</a> <font color="Maroon" title="c2">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9" title="CONLAT_1:func.9">B-meet</a> <font color="Maroon" title="c2">C</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="conlat_1.html#D23" title="CONLAT_1:def.23">CONLAT_1:def 23</a></i>;<br/>
<a NAME="E11:22"/><i><font color="Green" title="E19">A38</font></i>: 
for <font color="Olive" title="b1">a</font>, <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="c2">C</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">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="c2">C</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_5"/>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 ;<br/>

<a NAME="E2:22_5"/><b>then </b>
<font color="Maroon" title="c8">a</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="c9">b</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" title="CONLAT_1:th.47">CONLAT_1:47</a></i>;<br/>
<a NAME="E3:22_5"/><b>then </b><i><font color="Green" title="E20">A39</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="c8">a</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="c9">b</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" title="CONLAT_1:def.19">CONLAT_1:def 19</a></i>;<br/>
<a NAME="E4:22_5"/><i><font color="Green" title="E21">A40</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5_1"><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><br/>

<b>assume </b><a NAME="E1:22_5_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">o</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_5_1"/><i><font color="Green" title="E22">A41</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font> &amp; <font color="Maroon" title="c11">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 <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:22_5_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_5"><i><font color="Green" title="E20">A39</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_5_1"><i><font color="Green" title="E22">A41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c10">xa</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c11">xb</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E6:22_5"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>,<font color="Maroon" title="c10">xa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>,<font color="Maroon" title="c11">xb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> )
 
;<br/>
<a NAME="E7:22_5"/><b>then </b><i><font color="Green" title="E22">A42</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">xa</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xb</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E8:22_5"/>
( <font color="Maroon" title="c8">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c9">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )
 
<b>by </b><i><a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>;<br/>
<b>hence </b><a NAME="E9:22_5"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_5"><i><font color="Green" title="E21">A40</font></i></a>, <a class="txt" href="conlat_2.html#E7:22_5"><i><font color="Green" title="E22">A42</font></i></a>, <a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c8">fi</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b2">O</font>,<font color="Olive" title="b3">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Olive" title="b2">O</font> is   <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="c2">C</font>, <font color="Olive" title="b3">A</font> is   <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="c2">C</font> : ( <font color="Olive" title="b2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">o</font> where <font color="Olive" title="b4">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> </span>}</span>  &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">a</font> where <font color="Olive" title="b4">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">a</font> </span>}</span>  ) </span>}</span> ;<br/>
<a NAME="E12:22"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b2">O</font>,<font color="Olive" title="b3">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Olive" title="b2">O</font> is   <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="c2">C</font>, <font color="Olive" title="b3">A</font> is   <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="c2">C</font> : ( <font color="Olive" title="b2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">o</font> where <font color="Olive" title="b4">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> </span>}</span>  &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">a</font> where <font color="Olive" title="b4">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">a</font> </span>}</span>  ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</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><br/>

<b>assume </b><a NAME="E1:22_6"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b2">O</font>,<font color="Olive" title="b3">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Olive" title="b2">O</font> is   <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="c2">C</font>, <font color="Olive" title="b3">A</font> is   <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="c2">C</font> : ( <font color="Olive" title="b2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">o</font> where <font color="Olive" title="b4">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> </span>}</span>  &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">a</font> where <font color="Olive" title="b4">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">a</font> </span>}</span>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c11">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="c2">C</font>, <font color="Maroon" title="c12">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="c2">C</font><b> such that </b><br/><a NAME="E3:22_6"/><i><font color="Green" title="E20">A43</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span>  &amp; <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span>  )
 ;<br/>
<a NAME="E4:22_6"/><i><font color="Green" title="E21">A44</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_6_1"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">u</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><br/>

<b>assume </b><a NAME="E1:22_6_1_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 ;<br/>

<a NAME="E2:22_6_1_1"/><b>then </b>
<font color="Maroon" title="c13">u</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#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : for <font color="Olive" title="b2">a</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">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#U4" title="CONLAT_1:sel.4">Extent</a> of <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #) holds <br/><font color="Olive" title="b2">a</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">o</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D6" title="CONLAT_1:def.6">CONLAT_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">u'</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E4:22_6_1_1"/><i><font color="Green" title="E22">A45</font></i>: 
( <font color="Maroon" title="c14">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">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#U4" title="CONLAT_1:sel.4">Extent</a> of <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #) holds <br/><font color="Olive" title="b1">a</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c14">u'</font> ) )
 ;<br/>
<a NAME="E5:22_6_1_1"/><i><font color="Green" title="E23">A46</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span>  holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">v</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_1_1_1"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> 
 ;<br/>

<a NAME="E2:22_6_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">v</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c14">u'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a>, <a class="txt" href="conlat_2.html#E4:22_6_1_1"><i><font color="Green" title="E22">A45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_6_1_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6_1_1"/><i><font color="Green" title="E24">A47</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_6_1_1_2"/>
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">b</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_6_1_1_2"/><i><font color="Green" title="E25">A48</font></i>: 
( <font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> )
 ;<br/>
<a NAME="E4:22_6_1_1_2"/>
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_1_1_2"><i><font color="Green" title="E25">A48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:22_6_1_1_2"/>
<font color="Maroon" title="c15">q</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E5:22_6_1_1"><i><font color="Green" title="E23">A46</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_6_1_1_2"><i><font color="Green" title="E25">A48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_1_1"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c15">D</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g</font></span>)</span><b> such that </b><br/><a NAME="E2:22_6_1_1_3"/><i><font color="Green" title="E25">A49</font></i>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c15">D</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a>, <a class="ref" href="lattice6.html#D13" title="LATTICE6:def.13">LATTICE6:def 13</a></i>;<br/>
<a NAME="E3:22_6_1_1_3"/><i><font color="Green" title="E26">A50</font></i>: 
<font color="Maroon" title="c15">D</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_6_1_1_3"><i><font color="Green" title="E25">A49</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E4:22_6_1_1_3"/>
<font color="Maroon" title="c15">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">u</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><br/>

<b>assume </b><a NAME="E1:22_6_1_1_3_1"/><i><font color="Green" title="E27">A51</font></i>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:22_6_1_1_3_1"/><i><font color="Green" title="E28">A52</font></i>: 
( <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c17">v</font> as   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_1_1_3_1"><i><font color="Green" title="E28">A52</font></i></a></i>;<br/>
<a NAME="E5:22_6_1_1_3_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">v</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_1_1_3"><i><font color="Green" title="E26">A50</font></i></a>, <a class="txt" href="conlat_2.html#E1:22_6_1_1_3_1"><i><font color="Green" title="E27">A51</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_6_1_1_3_1"><i><font color="Green" title="E28">A52</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E6:22_6_1_1_3_1"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_1_1_3_1"><i><font color="Green" title="E28">A52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_6_1_1_3"/><b>then </b><i><font color="Green" title="E27">A53</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_6_1_1_3"><i><font color="Green" title="E25">A49</font></i></a>, <a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:22_6_1_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c10">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_6_1_1_3_2"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">v</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_6_1_1_3_2"/><i><font color="Green" title="E28">A54</font></i>: 
( <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">v</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">v</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_6_1_1_3_2"/>
<font color="Maroon" title="c16">u</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_1_1_3_2"><i><font color="Green" title="E28">A54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_1_1_3"/><b>then </b>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E8:22_6_1_1_3"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E5:22_6_1_1_3"><i><font color="Green" title="E27">A53</font></i></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_6_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E6:22_6_1_1"><i><font color="Green" title="E24">A47</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E9:22_6_1_1"/>
<font color="Maroon" title="c13">u</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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a>, <a class="txt" href="conlat_2.html#E4:22_6_1_1"><i><font color="Green" title="E22">A45</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c13">u</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><br/>

<b>assume </b><a NAME="E2:22_6_1"/>
<font color="Maroon" title="c13">u</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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">u'</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E4:22_6_1"/><i><font color="Green" title="E22">A55</font></i>: 
( <font color="Maroon" title="c14">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</font> &amp; <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a></i>;<br/>
<a NAME="E5:22_6_1"/><i><font color="Green" title="E23">A56</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span>  holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">v</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_1_2"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">v'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_6_1_2"/><i><font color="Green" title="E24">A57</font></i>: 
( <font color="Maroon" title="c16">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_6_1_2"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_6_1"><i><font color="Green" title="E22">A55</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_6_1_2"><i><font color="Green" title="E24">A57</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6_1"/>
for <font color="Olive" title="b1">v</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span>  holds <br/><font color="Olive" title="b1">v</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c14">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">v</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_1_3"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> </span>}</span> 
 ;<br/>

<a NAME="E2:22_6_1_3"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_6_1"><i><font color="Green" title="E23">A56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_6_1_3"/>
<font color="Maroon" title="c15">v</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c14">u'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_1"/><b>then </b>
<font color="Maroon" title="c14">u'</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#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : for <font color="Olive" title="b2">a</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">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#U4" title="CONLAT_1:sel.4">Extent</a> of <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #) holds <br/><font color="Olive" title="b2">a</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">o</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:22_6_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_6_1"><i><font color="Green" title="E22">A55</font></i></a>, <a class="ref" href="conlat_1.html#D6" title="CONLAT_1:def.6">CONLAT_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_6"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c2">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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_6_2"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c2">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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #) <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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">u</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><br/>

<b>assume </b><a NAME="E1:22_6_2_1"/>
<font color="Maroon" title="c13">u</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="c2">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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 ;<br/>

<a NAME="E2:22_6_2_1"/><b>then </b>
<font color="Maroon" title="c13">u</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="c2">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="c2">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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">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" title="CONLAT_1:def.7">CONLAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">u'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E4:22_6_2_1"/><i><font color="Green" title="E22">A58</font></i>: 
( <font color="Maroon" title="c14">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</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="c2">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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #) holds <br/><font color="Maroon" title="c14">u'</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="E5:22_6_2_1"/><i><font color="Green" title="E23">A59</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> </span>}</span>  holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">v</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_2_1_1"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 ;<br/>

<a NAME="E2:22_6_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">u'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c15">v</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a>, <a class="txt" href="conlat_2.html#E4:22_6_2_1"><i><font color="Green" title="E22">A58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_6_2_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6_2_1"/><i><font color="Green" title="E24">A60</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_6_2_1_2"/>
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">b</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_6_2_1_2"/><i><font color="Green" title="E25">A61</font></i>: 
( <font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font> &amp; <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font> )
 ;<br/>
<a NAME="E4:22_6_2_1_2"/>
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_2_1_2"><i><font color="Green" title="E25">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:22_6_2_1_2"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c15">q</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E5:22_6_2_1"><i><font color="Green" title="E23">A59</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_6_2_1_2"><i><font color="Green" title="E25">A61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_2_1"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c15">D</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">d</font></span>)</span><b> such that </b><br/><a NAME="E2:22_6_2_1_3"/><i><font color="Green" title="E25">A62</font></i>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c15">D</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a>, <a class="ref" href="lattice6.html#D14" title="LATTICE6:def.14">LATTICE6:def 14</a></i>;<br/>
<a NAME="E3:22_6_2_1_3"/><i><font color="Green" title="E26">A63</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c15">D</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_6_2_1_3"><i><font color="Green" title="E25">A62</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<a NAME="E4:22_6_2_1_3"/>
<font color="Maroon" title="c15">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">u</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><br/>

<b>assume </b><a NAME="E1:22_6_2_1_3_1"/><i><font color="Green" title="E27">A64</font></i>: 
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:22_6_2_1_3_1"/><i><font color="Green" title="E28">A65</font></i>: 
( <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c17">v</font> as   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_2_1_3_1"><i><font color="Green" title="E28">A65</font></i></a></i>;<br/>
<a NAME="E5:22_6_2_1_3_1"/>
<font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">v</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_2_1_3"><i><font color="Green" title="E26">A63</font></i></a>, <a class="txt" href="conlat_2.html#E1:22_6_2_1_3_1"><i><font color="Green" title="E27">A64</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_6_2_1_3_1"><i><font color="Green" title="E28">A65</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E6:22_6_2_1_3_1"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_2_1_3_1"><i><font color="Green" title="E28">A65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_6_2_1_3"/><b>then </b><i><font color="Green" title="E27">A66</font></i>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_6_2_1_3"><i><font color="Green" title="E25">A62</font></i></a>, <a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:22_6_2_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_6_2_1_3_2"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">v</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_6_2_1_3_2"/><i><font color="Green" title="E28">A67</font></i>: 
( <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">v</font> &amp; <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">v</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_6_2_1_3_2"/>
<font color="Maroon" title="c10">x</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c16">u</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6_2_1_3_2"><i><font color="Green" title="E28">A67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_2_1_3"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T40" title="LATTICE3:th.40">LATTICE3:40</a></i>;<br/>
<b>hence </b><a NAME="E8:22_6_2_1_3"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E5:22_6_2_1_3"><i><font color="Green" title="E27">A66</font></i></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22_6_2_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E6:22_6_2_1"><i><font color="Green" title="E24">A60</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E9:22_6_2_1"/>
<font color="Maroon" title="c13">u</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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a>, <a class="txt" href="conlat_2.html#E4:22_6_2_1"><i><font color="Green" title="E22">A58</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c13">u</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><br/>

<b>assume </b><a NAME="E2:22_6_2"/>
<font color="Maroon" title="c13">u</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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">u'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E4:22_6_2"/><i><font color="Green" title="E22">A68</font></i>: 
( <font color="Maroon" title="c14">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a></i>;<br/>
<a NAME="E5:22_6_2"/><i><font color="Green" title="E23">A69</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> </span>}</span>  holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">v</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_2_2"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">v'</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_6_2_2"/><i><font color="Green" title="E24">A70</font></i>: 
( <font color="Maroon" title="c16">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_6_2_2"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_6_2"><i><font color="Green" title="E22">A68</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_6_2_2"><i><font color="Green" title="E24">A70</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6_2"/>
for <font color="Olive" title="b1">v</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> </span>}</span>  holds <br/><font color="Maroon" title="c14">u'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">v</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_6_2_3"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 ;<br/>

<a NAME="E2:22_6_2_3"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_6_2"><i><font color="Green" title="E23">A69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_6_2_3"/>
<font color="Maroon" title="c14">u'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_6_2"/><b>then </b>
<font color="Maroon" title="c14">u'</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="c2">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="c2">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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">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="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:22_6_2"/>
<font color="Maroon" title="c13">u</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="c2">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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #)
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_6_2"><i><font color="Green" title="E22">A68</font></i></a>, <a class="ref" href="conlat_1.html#D7" title="CONLAT_1:def.7">CONLAT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_6"/><b>then </b>
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:22_6"><i><font color="Green" title="E21">A44</font></i></a>, <a class="txt" href="conlat_2.html#E21"><i><font color="Green" title="E13">Lm3</font></i></a>, <a class="ref" href="conlat_1.html#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></i>;<br/>
<a NAME="E7:22_6"/><b>then </b>
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c11">O</font>,<font color="Maroon" title="c12">A</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22"><i><font color="Green" title="E16">A29</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E8:22_6"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_6"><i><font color="Green" title="E20">A43</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">fi</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Olive" title="b2">O</font>,<font color="Olive" title="b3">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Olive" title="b2">O</font> is   <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="c2">C</font>, <font color="Olive" title="b3">A</font> is   <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="c2">C</font> : ( <font color="Olive" title="b2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">o</font> where <font color="Olive" title="b4">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> </span>}</span>  &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">a</font> where <font color="Olive" title="b4">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">a</font> </span>}</span>  ) </span>}</span>  as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E14:22"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">fi</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">y</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><br/>

<b>assume </b><a NAME="E1:22_7"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>set </b><font color="Maroon" title="c12">O</font> = <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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">y</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c13">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c11">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span> ;<br/>
<a NAME="E3:22_7"/>
<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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">y</font> </span>}</span>  <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="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">u</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><br/>

<b>assume </b><a NAME="E1:22_7_1"/>
<font color="Maroon" title="c14">u</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">y</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">u'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_7_1"/><i><font color="Green" title="E20">A71</font></i>: 
( <font color="Maroon" title="c15">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">y</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_7_1"/>
<font color="Maroon" title="c14">u</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="c2">C</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_7_1"><i><font color="Green" title="E20">A71</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">O</font> = <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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">y</font> </span>}</span>  as   <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="c2">C</font> ;<br/>
<a NAME="E5:22_7"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c11">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</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="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">u</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><br/>

<b>assume </b><a NAME="E1:22_7_2"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c11">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">u'</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_7_2"/><i><font color="Green" title="E20">A72</font></i>: 
( <font color="Maroon" title="c16">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">u</font> &amp; <font color="Maroon" title="c11">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">u'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_7_2"/>
<font color="Maroon" title="c15">u</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="c2">C</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_7_2"><i><font color="Green" title="E20">A72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c11">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span>  as   <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="c2">C</font> ;<br/>
<a NAME="E7:22_7"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c15">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">fi</font>
 
;<br/>
<b>hence </b><a NAME="E8:22_7"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">fi</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:22"/><b>then </b><i><font color="Green" title="E20">A73</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">fi</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E16:22"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">fi</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">fi</font> holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y1</font>, <font color="Maroon" title="c12">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_8"/><i><font color="Green" title="E21">A74</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">fi</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">fi</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c14">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="c2">C</font>, <font color="Maroon" title="c15">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="c2">C</font><b> such that </b><br/><a NAME="E3:22_8"/><i><font color="Green" title="E22">A75</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">z</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c15">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">z</font> </span>}</span>  &amp; <font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c13">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c16">z'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c17">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="c2">C</font>, <font color="Maroon" title="c18">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="c2">C</font><b> such that </b><br/><a NAME="E5:22_8"/><i><font color="Green" title="E23">A76</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z'</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c17">O'</font>,<font color="Maroon" title="c18">A'</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">O'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c16">z'</font> </span>}</span>  &amp; <font color="Maroon" title="c18">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c16">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="conlat_2.html#E1:22_8"><i><font color="Green" title="E21">A74</font></i></a></i>;<br/>
<font color="Maroon" title="c13">z</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_8"><i><font color="Green" title="E22">A75</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c16">z'</font>
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_8"><i><font color="Green" title="E23">A76</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c11">y1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_8"><i><font color="Green" title="E22">A75</font></i></a>, <a class="txt" href="conlat_2.html#E5:22_8"><i><font color="Green" title="E23">A76</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c12">y2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">fi</font> = <font color="Maroon" title="c9">fi</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span> <b>by </b><i><a class="txt" href="conlat_2.html#E15:22"><i><font color="Green" title="E20">A73</font></i></a>, <a class="ref" href="funct_1.html#D1" title="FUNCT_1:def.1">FUNCT_1:def 1</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:22"/><i><font color="Green" title="E21">A77</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:22_9"/><i><font color="Green" title="E22">A78</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">b</font>
 ;<br/>

<a NAME="E2:22_9"/><i><font color="Green" title="E23">A79</font></i>: 
<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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">a</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">b</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_9_1"><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_9_1"/>
<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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">a</font> </span>}</span> 
 ;<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="c2">C</font><b> such that </b><br/><a NAME="E3:22_9_1"/><i><font color="Green" title="E24">A80</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; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">a</font> )
 ;<br/>
<a NAME="E4:22_9_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">b</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E1:22_9"><i><font color="Green" title="E22">A78</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_9_1"><i><font color="Green" title="E24">A80</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E5:22_9_1"/>
<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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">b</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_9_1"><i><font color="Green" title="E24">A80</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:22_9"/><i><font color="Green" title="E24">A81</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">ya</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:22_9"/><i><font color="Green" title="E25">A82</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c13">ya</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">fi</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">xa</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c15">Oa</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="c2">C</font>, <font color="Maroon" title="c16">Aa</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="c2">C</font><b> such that </b><br/><a NAME="E7:22_9"/><i><font color="Green" title="E26">A83</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c13">ya</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">xa</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c15">Oa</font>,<font color="Maroon" title="c16">Aa</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">Oa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">xa</font> </span>}</span>  &amp; <font color="Maroon" title="c16">Aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c14">xa</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="conlat_2.html#E5:22_9"><i><font color="Green" title="E25">A82</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">yb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:22_9"/><i><font color="Green" title="E27">A84</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c17">yb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">fi</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_9"><i><font color="Green" title="E24">A81</font></i></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">xb</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c19">Ob</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="c2">C</font>, <font color="Maroon" title="c20">Ab</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="c2">C</font><b> such that </b><br/><a NAME="E11:22_9"/><i><font color="Green" title="E28">A85</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c17">yb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">xb</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c19">Ob</font>,<font color="Maroon" title="c20">Ab</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">Ob</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c18">xb</font> </span>}</span>  &amp; <font color="Maroon" title="c20">Ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c18">xb</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="conlat_2.html#E9:22_9"><i><font color="Green" title="E27">A84</font></i></a></i>;<br/>
<i><font color="Green" title="E29">A86</font></i>: <font color="Maroon" title="c11">a</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">xa</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c15">Oa</font>,<font color="Maroon" title="c16">Aa</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E7:22_9"><i><font color="Green" title="E26">A83</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c14">xa</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>
<i><font color="Green" title="E30">A87</font></i>: <font color="Maroon" title="c12">b</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">xb</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c19">Ob</font>,<font color="Maroon" title="c20">Ab</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E11:22_9"><i><font color="Green" title="E28">A85</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c18">xb</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E14:22_9"/><i><font color="Green" title="E31">A88</font></i>: 
<font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c15">Oa</font>,<font color="Maroon" title="c16">Aa</font> #)
 
<b>by </b><i><a class="txt" href="conlat_2.html#E5:22_9"><i><font color="Green" title="E25">A82</font></i></a>, <a class="txt" href="conlat_2.html#E7:22_9"><i><font color="Green" title="E26">A83</font></i></a>, <a class="txt" href="conlat_2.html#E12:22_9"><i><font color="Green" title="E29">A86</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E15:22_9"/><i><font color="Green" title="E32">A89</font></i>: 
<font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c19">Ob</font>,<font color="Maroon" title="c20">Ab</font> #)
 
<b>by </b><i><a class="txt" href="conlat_2.html#E9:22_9"><i><font color="Green" title="E27">A84</font></i></a>, <a class="txt" href="conlat_2.html#E11:22_9"><i><font color="Green" title="E28">A85</font></i></a>, <a class="txt" href="conlat_2.html#E13:22_9"><i><font color="Green" title="E30">A87</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c21">ca</font> = <font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font>;<br/>
<b>set </b><font color="Maroon" title="c22">cb</font> = <font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font>;<br/>
<a NAME="E16:22_9"/>
( the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Oa</font> &amp; the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Ob</font> )
 
<b>by </b><i><a class="txt" href="conlat_2.html#E14:22_9"><i><font color="Green" title="E31">A88</font></i></a>, <a class="txt" href="conlat_2.html#E15:22_9"><i><font color="Green" title="E32">A89</font></i></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>;<br/>
<a NAME="E17:22_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_9"><i><font color="Green" title="E23">A79</font></i></a>, <a class="txt" href="conlat_2.html#E7:22_9"><i><font color="Green" title="E26">A83</font></i></a>, <a class="txt" href="conlat_2.html#E11:22_9"><i><font color="Green" title="E28">A85</font></i></a>, <a class="txt" href="conlat_2.html#E12:22_9"><i><font color="Green" title="E29">A86</font></i></a>, <a class="txt" href="conlat_2.html#E13:22_9"><i><font color="Green" title="E30">A87</font></i></a>, <a class="ref" href="conlat_1.html#D19" title="CONLAT_1:def.19">CONLAT_1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E18:22_9"/>
<font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#T47" title="CONLAT_1:th.47">CONLAT_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:22"/><i><font color="Green" title="E22">A90</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c11">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>set </b><font color="Maroon" title="c13">O</font> = <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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span> ;<br/>
<a NAME="E2:22_10"/>
<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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  <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="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">u</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><br/>

<b>assume </b><a NAME="E1:22_10_1"/>
<font color="Maroon" title="c14">u</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">u'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_10_1"/><i><font color="Green" title="E23">A91</font></i>: 
( <font color="Maroon" title="c15">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_10_1"/>
<font color="Maroon" title="c14">u</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="c2">C</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_10_1"><i><font color="Green" title="E23">A91</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">O</font> = <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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  as   <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="c2">C</font> ;<br/>
<b>set </b><font color="Maroon" title="c15">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c12">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ;<br/>
<a NAME="E4:22_10"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c12">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</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="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">u</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><br/>

<b>assume </b><a NAME="E1:22_10_2"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c12">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">u'</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_10_2"/><i><font color="Green" title="E23">A92</font></i>: 
( <font color="Maroon" title="c17">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">u</font> &amp; <font color="Maroon" title="c12">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_10_2"/>
<font color="Maroon" title="c16">u</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="c2">C</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_10_2"><i><font color="Green" title="E23">A92</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c12">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span>  as   <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="c2">C</font> ;<br/>
<a NAME="E6:22_10"/><i><font color="Green" title="E23">A93</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">fi</font>
 
;<br/>
<b>set </b><font color="Maroon" title="c17">b</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #) </span>}</span> ,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E7:22_10"/>
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">fi</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E6:22_10"><i><font color="Green" title="E23">A93</font></i></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E8:22_10"/><b>then </b>
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E10:22"><i><font color="Green" title="E18">A37</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<a NAME="E9:22_10"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #),<span class="p2">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #) </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font>
 
;<br/>
<a NAME="E10:22_10"/><b>then </b><i><font color="Green" title="E24">A94</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #) </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E11:22_10"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">o'</font> where <font color="Olive" title="b2">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c18">D</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g</font></span>)</span><b> such that </b><br/><a NAME="E2:22_10_3"/><i><font color="Green" title="E25">A95</font></i>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c18">D</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a>, <a class="ref" href="lattice6.html#D13" title="LATTICE6:def.13">LATTICE6:def 13</a></i>;<br/>
<a NAME="E3:22_10_3"/><i><font color="Green" title="E26">A96</font></i>: 
<font color="Maroon" title="c18">D</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_10_3"><i><font color="Green" title="E25">A95</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E4:22_10_3"/>
<font color="Maroon" title="c18">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">o'</font> where <font color="Olive" title="b2">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">u</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><br/>

<b>assume </b><a NAME="E1:22_10_3_1"/><i><font color="Green" title="E27">A97</font></i>: 
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:22_10_3_1"/><i><font color="Green" title="E28">A98</font></i>: 
( <font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">v</font> = <font color="Maroon" title="c20">v</font> as   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_10_3_1"><i><font color="Green" title="E28">A98</font></i></a></i>;<br/>
<a NAME="E5:22_10_3_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">v</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_10_3"><i><font color="Green" title="E26">A96</font></i></a>, <a class="txt" href="conlat_2.html#E1:22_10_3_1"><i><font color="Green" title="E27">A97</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_10_3_1"><i><font color="Green" title="E28">A98</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E6:22_10_3_1"/><b>then </b>
<font color="Maroon" title="c21">v</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E7:22_10_3_1"/>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">o'</font> where <font color="Olive" title="b2">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_10_3_1"><i><font color="Green" title="E28">A98</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_10_3"/><b>then </b><i><font color="Green" title="E27">A99</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">o'</font> where <font color="Olive" title="b2">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_10_3"><i><font color="Green" title="E25">A95</font></i></a>, <a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:22_10_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">o'</font> where <font color="Olive" title="b2">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c12">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_10_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_10_3_2"/>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">o'</font> where <font color="Olive" title="b2">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">v</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_10_3_2"/><i><font color="Green" title="E28">A100</font></i>: 
( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">v</font> &amp; <font color="Maroon" title="c20">v</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c21">ov</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E5:22_10_3_2"/><i><font color="Green" title="E29">A101</font></i>: 
( <font color="Maroon" title="c21">ov</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">ov</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_10_3_2"><i><font color="Green" title="E28">A100</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:22_10_3_2"/>
<font color="Maroon" title="c19">u</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_10_3_2"><i><font color="Green" title="E28">A100</font></i></a>, <a class="txt" href="conlat_2.html#E5:22_10_3_2"><i><font color="Green" title="E29">A101</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_10_3"/><b>then </b>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">o'</font> where <font color="Olive" title="b2">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E8:22_10_3"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">o'</font> where <font color="Olive" title="b2">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">a</font> </span>}</span>  </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E5:22_10_3"><i><font color="Green" title="E27">A99</font></i></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:22_10"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E6:22_10"><i><font color="Green" title="E23">A93</font></i></a>, <a class="txt" href="conlat_2.html#E10:22_10"><i><font color="Green" title="E24">A94</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:22"/><i><font color="Green" title="E23">A102</font></i>: 
for <font color="Olive" title="b1">x</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="c2">C</font></span>)</span> holds  <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</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="c2">C</font></span>)</span>;<br/>

<b>set </b><font color="Maroon" title="c12">y</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:22_11"/><i><font color="Green" title="E24">A103</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>,<span class="p2">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font>
 
;<br/>
<i><font color="Green" title="E25">A104</font></i>: <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>
<br/>.= 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="conlat_2.html#E1:22_11"><i><font color="Green" title="E24">A103</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>
;<br/>
<a NAME="E3:22_11"/><i><font color="Green" title="E26">A105</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="c2">C</font>  st <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/>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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_2"><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="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_11_2"/><i><font color="Green" title="E27">A106</font></i>: 
<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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c14">a'</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E2:22_11_2"/>
<font color="Maroon" title="c14">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ;<br/>

<a NAME="E3:22_11_2"/><b>then </b>
<font color="Maroon" title="c14">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c11">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#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></i>;<br/>
<a NAME="E4:22_11_2"/><b>then </b>
<font color="Maroon" title="c14">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : for <font color="Olive" title="b2">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Olive" title="b2">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D6" title="CONLAT_1:def.6">CONLAT_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">aa</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E6:22_11_2"/><i><font color="Green" title="E28">A107</font></i>: 
( <font color="Maroon" title="c15">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a'</font> &amp; ( 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="c2">C</font>  st <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> 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="Maroon" title="c15">aa</font> ) )
 ;<br/>
<a NAME="E7:22_11_2"/>
<font color="Maroon" title="c13">o'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c14">a'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E1:22_11_2"><i><font color="Green" title="E27">A106</font></i></a>, <a class="txt" href="conlat_2.html#E6:22_11_2"><i><font color="Green" title="E28">A107</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:22_11_2"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:22_11"/><i><font color="Green" title="E27">A108</font></i>: 
for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font></span>)</span> where <font color="Olive" title="b2">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">o</font></span>)</span> where <font color="Olive" title="b3">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b3">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span>  holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_11_3"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">u</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_11_3"/><i><font color="Green" title="E28">A109</font></i>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font> &amp;  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font> )
 ;<br/>
<a NAME="E4:22_11_3"/><i><font color="Green" title="E29">A110</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="c2">C</font>  st <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">o</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_11_3_1"/>
<font color="Maroon" title="c15">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_11_3_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o'</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E3:22_11_3_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E4:22_11_3_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_11_3"><i><font color="Green" title="E28">A109</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_11_3"/>
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="c2">C</font>  st <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> 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="Maroon" title="c14">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">o</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_11_3_2"/>
<font color="Maroon" title="c15">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_11_3_2"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:22_11_3"><i><font color="Green" title="E29">A110</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_11_3_2"/>
<font color="Maroon" title="c15">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c14">u</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_11_3"/><b>then </b>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : for <font color="Olive" title="b2">o'</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Olive" title="b2">o'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 
;<br/>
<a NAME="E7:22_11_3"/><b>then </b>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c11">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#D6" title="CONLAT_1:def.6">CONLAT_1:def 6</a></i>;<br/>
<a NAME="E8:22_11_3"/><b>then </b>
<font color="Maroon" title="c14">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">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#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E9:22_11_3"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_11_3"><i><font color="Green" title="E28">A109</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_11"/><i><font color="Green" title="E28">A111</font></i>: 
for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span>  holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font></span>)</span> where <font color="Olive" title="b2">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">o</font></span>)</span> where <font color="Olive" title="b3">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b3">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_11_4"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">b</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_11_4"/><i><font color="Green" title="E29">A112</font></i>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">b</font> &amp; <font color="Maroon" title="c14">b</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> )
 ;<br/>
<a NAME="E4:22_11_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o'</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:22_11_4_1"/>
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o'</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">u</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_11_4_1"/><i><font color="Green" title="E30">A113</font></i>: 
( <font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">u</font> &amp; <font color="Maroon" title="c16">u</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="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:22_11_4_1"/>
<font color="Maroon" title="c15">q</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_11"><i><font color="Green" title="E26">A105</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_11_4"><i><font color="Green" title="E29">A112</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_11_4_1"><i><font color="Green" title="E30">A113</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_11_4"/><b>then </b>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">b</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E6:22_11_4"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_11_4"><i><font color="Green" title="E29">A112</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_11"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c13">D</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">d</font></span>)</span><b> such that </b><br/><a NAME="E2:22_11_5"/><i><font color="Green" title="E29">A114</font></i>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c13">D</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a>, <a class="ref" href="lattice6.html#D14" title="LATTICE6:def.14">LATTICE6:def 14</a></i>;<br/>
<a NAME="E3:22_11_5"/><i><font color="Green" title="E30">A115</font></i>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c13">D</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_11_5"><i><font color="Green" title="E29">A114</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<a NAME="E4:22_11_5"/>
<font color="Maroon" title="c13">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">u</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><br/>

<b>assume </b><a NAME="E1:22_11_5_1"/><i><font color="Green" title="E31">A116</font></i>: 
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:22_11_5_1"/><i><font color="Green" title="E32">A117</font></i>: 
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c15">v</font> as   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_11_5_1"><i><font color="Green" title="E32">A117</font></i></a></i>;<br/>
<a NAME="E5:22_11_5_1"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">v</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_11_5"><i><font color="Green" title="E30">A115</font></i></a>, <a class="txt" href="conlat_2.html#E1:22_11_5_1"><i><font color="Green" title="E31">A116</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_11_5_1"><i><font color="Green" title="E32">A117</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E6:22_11_5_1"/>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_11_5_1"><i><font color="Green" title="E32">A117</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_11_5"/><b>then </b><i><font color="Green" title="E31">A118</font></i>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:22_11_5"><i><font color="Green" title="E29">A114</font></i></a>, <a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:22_11_5"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_11_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_11_5_2"/>
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">v</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_11_5_2"/><i><font color="Green" title="E32">A119</font></i>: 
( <font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> &amp;  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_11_5_2"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c14">u</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_11_5_2"><i><font color="Green" title="E32">A119</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:22_11_5"/><b>then </b>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T40" title="LATTICE3:th.40">LATTICE3:40</a></i>;<br/>
<b>hence </b><a NAME="E8:22_11_5"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font></span>)</span> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">o</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span> 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="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E5:22_11_5"><i><font color="Green" title="E31">A118</font></i></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:22_11"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22_11"><i><font color="Green" title="E25">A104</font></i></a>, <a class="txt" href="conlat_2.html#E4:22_11"><i><font color="Green" title="E27">A108</font></i></a>, <a class="txt" href="conlat_2.html#E5:22_11"><i><font color="Green" title="E28">A111</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:22"/><i><font color="Green" title="E24">A120</font></i>: 
for <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="c2">C</font></span>)</span> holds  <font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</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="c2">C</font></span>)</span>;<br/>

<a NAME="E1:22_12"/><i><font color="Green" title="E25">A121</font></i>: 
<font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E20:22"><i><font color="Green" title="E23">A102</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">x</font> =  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c13">O</font> = <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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> </span>}</span> ;<br/>
<a NAME="E2:22_12"/>
<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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> </span>}</span>  <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="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">u</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><br/>

<b>assume </b><a NAME="E1:22_12_1"/>
<font color="Maroon" title="c14">u</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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">u'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_12_1"/><i><font color="Green" title="E26">A122</font></i>: 
( <font color="Maroon" title="c15">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_12_1"/>
<font color="Maroon" title="c14">u</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="c2">C</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_12_1"><i><font color="Green" title="E26">A122</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">O</font> = <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="c2">C</font> : <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> </span>}</span>  as   <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="c2">C</font> ;<br/>
<b>set </b><font color="Maroon" title="c15">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> ;<br/>
<a NAME="E4:22_12"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</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="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">u</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><br/>

<b>assume </b><a NAME="E1:22_12_2"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">u'</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_12_2"/><i><font color="Green" title="E26">A123</font></i>: 
( <font color="Maroon" title="c17">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">u</font> &amp;  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">u'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:22_12_2"/>
<font color="Maroon" title="c16">u</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="c2">C</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_12_2"><i><font color="Green" title="E26">A123</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a'</font> where <font color="Olive" title="b1">a'</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> :  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b2">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b2">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> </span>}</span>  as   <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="c2">C</font> ;<br/>
<a NAME="E6:22_12"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span>,<a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">fi</font>
 
;<br/>
<a NAME="E7:22_12"/><b>then </b><i><font color="Green" title="E26">A124</font></i>: 
<font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #)
 
<b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E8:22_12"/><b>then </b><i><font color="Green" title="E27">A125</font></i>: 
 <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c14">O</font>,<font color="Maroon" title="c16">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E10:22"><i><font color="Green" title="E18">A37</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<a NAME="E9:22_12"/><i><font color="Green" title="E28">A126</font></i>: 
<font color="Maroon" title="c14">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_12_3"/>
<font color="Maroon" title="c14">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">u</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><br/>

<b>assume </b><a NAME="E1:22_12_3_1"/>
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">O</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">o'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_12_3_1"/><i><font color="Green" title="E29">A127</font></i>: 
( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o'</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> )
 ;<br/>
<a NAME="E4:22_12_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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">a'</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_12_3_1_1"/>
<font color="Maroon" title="c19">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_12_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E3:22_12_3_1_1"/><b>then </b>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">a'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E4:22_12_3_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">a'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_12_3_1"><i><font color="Green" title="E29">A127</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_12_3_1"/>
<font color="Maroon" title="c17">u</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:22_12_3_1"><i><font color="Green" title="E29">A127</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c17">u</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><br/>

<b>assume </b><a NAME="E2:22_12_3"/>
<font color="Maroon" title="c17">u</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">o'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E4:22_12_3"/><i><font color="Green" title="E29">A128</font></i>: 
( <font color="Maroon" title="c18">o'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> ) )
 ;<br/>
<a NAME="E5:22_12_3"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">o'</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:22_12_3_2"/>
<font color="Maroon" title="c19">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">qa</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_12_3_2"/><i><font color="Green" title="E30">A129</font></i>: 
( <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">qa</font> &amp; <font color="Maroon" title="c20">qa</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:22_12_3_2"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">o'</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c19">q</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_12_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="txt" href="conlat_2.html#E3:22_12_3_2"><i><font color="Green" title="E30">A129</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_12_3"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">o'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font> : <font color="Olive" title="b1">u</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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E7:22_12_3"/>
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">O</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_12_3"><i><font color="Green" title="E29">A128</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:22_12"/>
<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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> 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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:22_12_4"/>
<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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> 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> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">u</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><br/>

<b>assume </b><a NAME="E1:22_12_4_1"/>
<font color="Maroon" title="c17">u</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">u'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:22_12_4_1"/><i><font color="Green" title="E29">A130</font></i>: 
( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">u'</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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font> ) )
 ;<br/>
<a NAME="E4:22_12_4_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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c18">u'</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="22_12_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">a'</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_12_4_1_1"/>
<font color="Maroon" title="c19">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_12_4_1_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">a'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E3:22_12_4_1"><i><font color="Green" title="E29">A130</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_12_4_1_1"/>
<font color="Maroon" title="c18">u'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c19">a'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_12_4_1"/>
<font color="Maroon" title="c17">u</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> 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="txt" href="conlat_2.html#E3:22_12_4_1"><i><font color="Green" title="E29">A130</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c17">u</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><br/>

<b>assume </b><a NAME="E2:22_12_4"/>
<font color="Maroon" title="c17">u</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> 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>then consider </b><font color="Maroon" title="c18">u'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E4:22_12_4"/><i><font color="Green" title="E29">A131</font></i>: 
( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">u'</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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c18">u'</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="E5:22_12_4"/>
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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_12_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">a'</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c2">C</font>;<br/>

<b>assume </b><a NAME="E1:22_12_4_2"/>
<font color="Maroon" title="c19">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:22_12_4_2"/><b>then </b>
<font color="Maroon" title="c18">u'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c19">a'</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:22_12_4"><i><font color="Green" title="E29">A131</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_12_4_2"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">a'</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:22"><i><font color="Green" title="E15">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_12_4"/>
<font color="Maroon" title="c17">u</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="c2">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="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E4:22_12_4"><i><font color="Green" title="E29">A131</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E29">A132</font></i>: <font color="Maroon" title="c14">O</font> = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c2">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 <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="conlat_2.html#E9:22_12"><i><font color="Green" title="E28">A126</font></i></a>, <a class="ref" href="conlat_1.html#D7" title="CONLAT_1:def.7">CONLAT_1:def 7</a></i>
<br/>.= 
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="c11">a</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#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c16">A</font> = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c11">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="conlat_2.html#E8:22_12"><i><font color="Green" title="E27">A125</font></i></a>, <a class="ref" href="conlat_1.html#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></i>
<br/>.= 
the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</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#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E13:22_12"/>
<font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E1:22_12"><i><font color="Green" title="E25">A121</font></i></a>, <a class="txt" href="conlat_2.html#E7:22_12"><i><font color="Green" title="E26">A124</font></i></a>, <a class="txt" href="conlat_2.html#E11:22_12"><i><font color="Green" title="E29">A132</font></i></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:22"/><i><font color="Green" title="E25">A133</font></i>: 
for <font color="Olive" title="b1">a</font>, <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="c2">C</font></span>)</span>  st <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">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="c2">C</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_13"/><i><font color="Green" title="E26">A134</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font>
 ;<br/>

<a NAME="E2:22_13"/>
( <font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b</font> )
 
<b>by </b><i><a class="txt" href="conlat_2.html#E21:22"><i><font color="Green" title="E24">A120</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:22_13"/>
<font color="Maroon" title="c11">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E18:22"><i><font color="Green" title="E21">A77</font></i></a>, <a class="txt" href="conlat_2.html#E1:22_13"><i><font color="Green" title="E26">A134</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:22"/><b>then </b><i><font color="Green" title="E26">A135</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E19"><i><font color="Green" title="E11">Lm1</font></i></a></i>;<br/>
<a NAME="E24:22"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">u</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><br/>

<b>assume </b><a NAME="E1:22_14"/><i><font color="Green" title="E27">A136</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E2:22_14"/><b>then </b>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">fi</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:22_14"/><i><font color="Green" title="E28">A137</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c12">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">fi</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E5:22_14"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">fi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span> &amp; <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">fi</font> )
 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:22_14"><i><font color="Green" title="E28">A137</font></i></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">v</font> = <font color="Maroon" title="c12">v</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="c2">C</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="conlat_2.html#E1:22_14"><i><font color="Green" title="E27">A136</font></i></a></i>;<br/>
<i><font color="Green" title="E29">A138</font></i>: <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> = 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">u</font></span>)</span>
<b>by </b><i><a class="txt" href="conlat_2.html#E4:22_14"><i><font color="Green" title="E28">A137</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c14">u</font>
<b>by </b><i><a class="txt" href="conlat_2.html#E19:22"><i><font color="Green" title="E22">A90</font></i></a></i>
;<br/>
<a NAME="E9:22_14"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:22_14"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E8:22_14"><i><font color="Green" title="E29">A138</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:22"/><b>then </b><i><font color="Green" title="E27">A139</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c7">f</font> as    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="conlat_2.html#E11:22"><i><font color="Green" title="E19">A38</font></i></a>, <a class="txt" href="conlat_2.html#E22:22"><i><font color="Green" title="E25">A133</font></i></a>, <a class="txt" href="conlat_2.html#E23:22"><i><font color="Green" title="E26">A135</font></i></a>, <a class="txt" href="conlat_2.html#E20"><i><font color="Green" title="E12">Lm2</font></i></a></i>;<br/>
<a NAME="E27:22"/><i><font color="Green" title="E28">A140</font></i>: 
<font color="Maroon" title="c11">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E25:22"><i><font color="Green" title="E27">A139</font></i></a>, <a class="ref" href="lattice4.html#D3" title="LATTICE4:def.3">LATTICE4:def 3</a></i>;<br/>
<a NAME="E28:22"/>
<font color="Maroon" title="c11">f</font> is <a href="lattice4.html#V1" title="LATTICE4:attr.1">monomorphism</a>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E23:22"><i><font color="Green" title="E26">A135</font></i></a>, <a class="ref" href="lattice4.html#D2" title="LATTICE4:def.2">LATTICE4:def 2</a></i>;<br/>
<a NAME="E29:22"/><b>then </b>
<font color="Maroon" title="c11">f</font> is <a href="lattice4.html#V3" title="LATTICE4:attr.3">isomorphism</a>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E27:22"><i><font color="Green" title="E28">A140</font></i></a>, <a class="ref" href="lattice4.html#D4" title="LATTICE4:def.4">LATTICE4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E30:22"/>
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="lattice4.html#D5" title="LATTICE4:def.5">LATTICE4:def 5</a></i>;<br/>


</div>
