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

<b>set </b><font color="Maroon" title="c2">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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> :  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> </span>}</span> ;<br/>
<a NAME="E1:34_1_1"/><i><font color="Green" title="E20">A1</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</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="c1">C</font></span>)</span>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice2.html#D2" title="LATTICE2:def.2">LATTICE2:def 2</a></i>;<br/>
<a NAME="E2:34_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">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="c1">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="c1">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="c1">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="E3:34_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10" title="CONLAT_1:func.10">B-join</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9" title="CONLAT_1:func.9">B-meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></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="E4:34_1_1"/><i><font color="Green" title="E23">A4</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  <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="c1">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="c1">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="c1">C</font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="#E2:34_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="lattice2.html#D2" title="LATTICE2:def.2">LATTICE2:def 2</a></i>;<br/>
<a NAME="E5:34_1_1"/>
<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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> :  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</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"><span class="p3">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></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="34_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> :  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> </span>}</span>  or <font color="Maroon" title="c3">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"><span class="p3">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1_1"/>
<font color="Maroon" title="c3">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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> :  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">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"><span class="p3">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c4">O</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c5">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E3:34_1_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c3">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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c4">O</font>,<font color="Maroon" title="c5">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c4">O</font>,<font color="Maroon" title="c5">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c4">O</font>,<font color="Maroon" title="c5">A</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> )
 ;<br/>
<a NAME="E4:34_1_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
 <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c4">O</font>,<font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:34_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E3:34_1_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<a NAME="E5:34_1_1_1"/>
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c4">O</font>,<font color="Maroon" title="c5">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a>  is  <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> 
 
<b>by </b><i><a class="txt" href="#E3:34_1_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="conlat_2.html#E32"><i><font color="Green" title="E19">Lm4</font></i></a></i>;<br/>
<a NAME="E6:34_1_1_1"/><b>then </b>
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c4">O</font>,<font color="Maroon" title="c5">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a>  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:34_1_1"><i><font color="Green" title="E22">A3</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="E7:34_1_1_1"/>
<font color="Maroon" title="c3">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"><span class="p3">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E3:34_1_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E4:34_1_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> :  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></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="E7:34_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </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="c3">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> or <font color="Maroon" title="c4">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="c3">f</font> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1_2"/>
<font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">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="c3">f</font></span><br/>

<a NAME="E2:34_1_1_2"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c4">y</font> is  <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="#E4:34_1_1"><i><font color="Green" title="E23">A4</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="c5">O'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c6">A'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:34_1_1_2"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">O'</font>,<font color="Maroon" title="c6">A'</font> #)
 ;<br/>
<b>set </b><font color="Maroon" title="c7">z</font> = <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">O'</font>,<font color="Maroon" title="c6">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> ;<br/>
<a NAME="E5:34_1_1_2"/>
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">O'</font>,<font color="Maroon" title="c6">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a>  is  <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> 
 
<b>by </b><i><a class="txt" href="#E2:34_1_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="#E4:34_1_1_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="conlat_2.html#E32"><i><font color="Green" title="E19">Lm4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">z</font> = <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">O'</font>,<font color="Maroon" title="c6">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a>  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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E3:34_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<a NAME="E7:34_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c8">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="c3">f</font>
 
<b>by </b><i><a class="txt" href="#E2:34_1_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="#E4:34_1_1_2"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:34_1_1_2"/>
<font color="Maroon" title="c4">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="c3">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:34_1_1"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </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="c3">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="E9:34_1_1"/>
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="c3">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="c3">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="34_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="c3">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">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="c3">f</font> implies <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1_3"/><i><font color="Green" title="E25">A10</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="c3">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">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="c3">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font></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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E3:34_1_1_3"/><i><font color="Green" title="E26">A11</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c8">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c8">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">O'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c10">A'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E5:34_1_1_3"/><i><font color="Green" title="E27">A12</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c9">O'</font>,<font color="Maroon" title="c10">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c9">O'</font>,<font color="Maroon" title="c10">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c9">O'</font>,<font color="Maroon" title="c10">A'</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E1:34_1_1_3"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/>
 <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c7">O</font>,<font color="Maroon" title="c8">A</font> #) = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="#E3:34_1_1_3"><i><font color="Green" title="E26">A11</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="c4">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="c4">x</font>,<font color="Maroon" title="c6">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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c9">O'</font>,<font color="Maroon" title="c10">A'</font> #)
<b>by </b><i><a class="txt" href="#E5:34_1_1_3"><i><font color="Green" title="E27">A12</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="c5">y1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">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="#E3:34_1_1_3"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="#E5:34_1_1_3"><i><font color="Green" title="E27">A12</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="c6">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/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E8:34_1_1"><i><font color="Green" title="E24">A9</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="E11:34_1_1"/><i><font color="Green" title="E25">A13</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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </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="c4">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="c4">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="34_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1_4"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="txt" href="#E1:34_1_1"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E3:34_1_1_4"/><i><font color="Green" title="E27">A15</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </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>then consider </b><font color="Maroon" title="c9">fa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:34_1_1_4"/><i><font color="Green" title="E28">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">fa</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="c4">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/>
<b>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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E7:34_1_1_4"/><i><font color="Green" title="E29">A17</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">fa</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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E5:34_1_1_4"><i><font color="Green" title="E28">A16</font></i></a></i>;<br/>
<i><font color="Green" title="E30">A18</font></i>: <font color="Maroon" title="c5">a</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E7:34_1_1_4"><i><font color="Green" title="E29">A17</font></i></a>, <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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</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="E9:34_1_1_4"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fa</font>
 
<b>by </b><i><a class="txt" href="#E5:34_1_1_4"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">fb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:34_1_1_4"/><i><font color="Green" title="E32">A20</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">fb</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="c4">f</font>
 <b>by </b><i><a class="txt" href="#E3:34_1_1_4"><i><font color="Green" title="E27">A15</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="c13">O'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E13:34_1_1_4"/><i><font color="Green" title="E33">A21</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">fb</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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E11:34_1_1_4"><i><font color="Green" title="E32">A20</font></i></a></i>;<br/>
<i><font color="Green" title="E34">A22</font></i>: <font color="Maroon" title="c6">b</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E13:34_1_1_4"><i><font color="Green" title="E33">A21</font></i></a>, <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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</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="E15:34_1_1_4"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fb</font>
 
<b>by </b><i><a class="txt" href="#E11:34_1_1_4"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E16:34_1_1_4"/>
<font color="Maroon" title="c5">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="txt" href="#E1:34_1_1_4"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/>
<a NAME="E17:34_1_1_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E18:34_1_1_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T20" title="LATTICE3:th.20">LATTICE3:20</a></i>;<br/>
<a NAME="E19:34_1_1_4"/><b>then </b><i><font color="Green" title="E36">A24</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">aa</font> = <font color="Maroon" title="c5">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> , <font color="Maroon" title="c16">bb</font> = <font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E1:34_1_1"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E21:34_1_1_4"/><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c16">bb</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">aa</font>
 
<b>by </b><i><a class="txt" href="#E19:34_1_1_4"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E22:34_1_1_4"/><i><font color="Green" title="E38">A26</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c15">aa</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">aa</font>
 
;<br/>
<a NAME="E23:34_1_1_4"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c16">bb</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">bb</font>
 
;<br/>
<a NAME="E24:34_1_1_4"/><b>then </b>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c16">bb</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c15">aa</font>
 
<b>by </b><i><a class="txt" href="#E21:34_1_1_4"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="#E22:34_1_1_4"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/>
<a NAME="E25:34_1_1_4"/><b>then </b><i><font color="Green" title="E39">A27</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c16">bb</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"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c15">aa</font></span>)</span> <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="E26:34_1_1_4"/>
<font color="Maroon" title="c9">fa</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">f</font>
 
<b>by </b><i><a class="txt" href="#E5:34_1_1_4"><i><font color="Green" title="E28">A16</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="c17">fa</font> = <font color="Maroon" title="c9">fa</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E28:34_1_1_4"/>
<font color="Maroon" title="c12">fb</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">f</font>
 
<b>by </b><i><a class="txt" href="#E11:34_1_1_4"><i><font color="Green" title="E32">A20</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="c18">fb</font> = <font color="Maroon" title="c12">fb</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> ;<br/>
<font color="Maroon" title="c17">fa</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E7:34_1_1_4"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E40">A28</font></i>: the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c17">fa</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</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/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #)
<b>by </b><i><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E8:34_1_1_4"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>
;<br/>
<font color="Maroon" title="c18">fb</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E13:34_1_1_4"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<b>then </b>the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">fb</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</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/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #)
<b>by </b><i><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E14:34_1_1_4"><i><font color="Green" title="E34">A22</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="E34:34_1_1_4"/><b>then </b>
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">fb</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#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c17">fa</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E25:34_1_1_4"><i><font color="Green" title="E39">A27</font></i></a>, <a class="txt" href="#E31:34_1_1_4"><i><font color="Green" title="E40">A28</font></i></a>, <a class="ref" href="conlat_1.html#D19" title="CONLAT_1:def.19">CONLAT_1:def 19</a></i>;<br/>
<a NAME="E35:34_1_1_4"/><b>then </b>
<font color="Maroon" title="c17">fa</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="conlat_1.html#R2" title="CONLAT_1:pred.2">is-SubConcept-of</a> <font color="Maroon" title="c18">fb</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#T31" title="CONLAT_1:th.31">CONLAT_1:31</a></i>;<br/>
<b>hence </b><a NAME="E36:34_1_1_4"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E9:34_1_1_4"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="#E15:34_1_1_4"><i><font color="Green" title="E35">A23</font></i></a>, <a class="ref" href="conlat_1.html#T47" title="CONLAT_1:th.47">CONLAT_1:47</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:34_1_1"/>
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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></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">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> or <font color="Maroon" title="c5">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="c4">f</font> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1_5"/>
<font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">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="c4">f</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c5">u</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">O'</font> = the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A'</font> = the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<b>set </b><font color="Maroon" title="c9">CP</font> =  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #);<br/>
<i><font color="Green" title="E26">A29</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="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #) = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_2.html#T18" target="_self" title="CONLAT_2:th.18">Th18</a></i>
<br/>.= 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #)
<b>by </b><i><a class="ref" href="conlat_1.html#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></i>
;<br/>
<i><font color="Green" title="E27">A30</font></i>: <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #) = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_2.html#T19" target="_self" title="CONLAT_2:th.19">Th19</a></i>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #)
<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="E7:34_1_1_5"/>
(  not the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="ref" href="conlat_1.html#D11" title="CONLAT_1:def.11">CONLAT_1:def 11</a></i>;<br/>
<a NAME="E8:34_1_1_5"/><b>then </b><i><font color="Green" title="E28">A31</font></i>: 
 <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="#E5:34_1_1_5"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="#E6:34_1_1_5"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="conlat_1.html#D11" title="CONLAT_1:def.11">CONLAT_1:def 11</a>, <a class="ref" href="conlat_1.html#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></i>;<br/>
<a NAME="E9:34_1_1_5"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </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="E10:34_1_1_5"/><b>then </b>
 <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</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">f</font>
 
<b>by </b><i><a class="txt" href="#E1:34_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E2:34_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E8:34_1_1_5"><i><font color="Green" title="E28">A31</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="c10">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:34_1_1_5"/><i><font color="Green" title="E29">A32</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #),<font color="Maroon" title="c10">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="c4">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/>
<b>consider </b><font color="Maroon" title="c11">Ov</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c12">Av</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E14:34_1_1_5"/><i><font color="Green" title="E30">A33</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #),<font color="Maroon" title="c10">v</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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E12:34_1_1_5"><i><font color="Green" title="E29">A32</font></i></a></i>;<br/>
<a NAME="E15:34_1_1_5"/><i><font color="Green" title="E31">A34</font></i>: 
<font color="Maroon" title="c10">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="c4">f</font>
 
<b>by </b><i><a class="txt" href="#E12:34_1_1_5"><i><font color="Green" title="E29">A32</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="c10">v</font> as  <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a>  <b>by </b><i><a class="txt" href="#E3:34_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<i><font color="Green" title="E32">A35</font></i>: <font color="Maroon" title="c13">v</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E14:34_1_1_5"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<i><font color="Green" title="E33">A36</font></i>:  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c7">O'</font> #) = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E14:34_1_1_5"><i><font color="Green" title="E30">A33</font></i></a>, <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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c11">Ov</font>,<font color="Maroon" title="c12">Av</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="E19:34_1_1_5"/><b>then </b><i><font color="Green" title="E34">A37</font></i>: 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E17:34_1_1_5"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>;<br/>
<a NAME="E20:34_1_1_5"/>
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E17:34_1_1_5"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="#E18:34_1_1_5"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E21:34_1_1_5"/>
<font color="Maroon" title="c5">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="c4">f</font>
 <b>by </b><i><a class="txt" href="#E15:34_1_1_5"><i><font color="Green" title="E31">A34</font></i></a>, <a class="txt" href="#E19:34_1_1_5"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:34_1_1"/><b>then </b><i><font color="Green" title="E26">A38</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span>
 
<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="E14:34_1_1"/><i><font color="Green" title="E27">A39</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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>  st <font color="Maroon" title="c4">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="c4">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="34_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b</font> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1_6"/><i><font color="Green" title="E28">A40</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="txt" href="#E1:34_1_1"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E3:34_1_1_6"/><i><font color="Green" title="E29">A41</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </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>then consider </b><font color="Maroon" title="c9">fa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:34_1_1_6"/><i><font color="Green" title="E30">A42</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">fa</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="c4">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/>
<b>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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E7:34_1_1_6"/><i><font color="Green" title="E31">A43</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">fa</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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E5:34_1_1_6"><i><font color="Green" title="E30">A42</font></i></a></i>;<br/>
<i><font color="Green" title="E32">A44</font></i>: <font color="Maroon" title="c5">a</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E7:34_1_1_6"><i><font color="Green" title="E31">A43</font></i></a>, <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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</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/>
<b>consider </b><font color="Maroon" title="c12">fb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:34_1_1_6"/><i><font color="Green" title="E33">A45</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">fb</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="c4">f</font>
 <b>by </b><i><a class="txt" href="#E3:34_1_1_6"><i><font color="Green" title="E29">A41</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="c13">O'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E12:34_1_1_6"/><i><font color="Green" title="E34">A46</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">fb</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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E10:34_1_1_6"><i><font color="Green" title="E33">A45</font></i></a></i>;<br/>
<i><font color="Green" title="E35">A47</font></i>: <font color="Maroon" title="c6">b</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E12:34_1_1_6"><i><font color="Green" title="E34">A46</font></i></a>, <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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</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="E14:34_1_1_6"/><i><font color="Green" title="E36">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="txt" href="#E1:34_1_1_6"><i><font color="Green" title="E28">A40</font></i></a>, <a class="ref" href="conlat_1.html#T47" title="CONLAT_1:th.47">CONLAT_1:47</a></i>;<br/>
<a NAME="E15:34_1_1_6"/>
<font color="Maroon" title="c9">fa</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">f</font>
 
<b>by </b><i><a class="txt" href="#E5:34_1_1_6"><i><font color="Green" title="E30">A42</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="c15">fa</font> = <font color="Maroon" title="c9">fa</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E17:34_1_1_6"/>
<font color="Maroon" title="c12">fb</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">f</font>
 
<b>by </b><i><a class="txt" href="#E10:34_1_1_6"><i><font color="Green" title="E33">A45</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="c16">fb</font> = <font color="Maroon" title="c12">fb</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> ;<br/>
<i><font color="Green" title="E37">A49</font></i>: <font color="Maroon" title="c15">fa</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E7:34_1_1_6"><i><font color="Green" title="E31">A43</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<i><font color="Green" title="E38">A50</font></i>: the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">fa</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E5:34_1_1_6"><i><font color="Green" title="E30">A42</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>
<br/>.= 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E19:34_1_1_6"><i><font color="Green" title="E37">A49</font></i></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #)
<b>by </b><i><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E8:34_1_1_6"><i><font color="Green" title="E32">A44</font></i></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>
;<br/>
<i><font color="Green" title="E39">A51</font></i>: <font color="Maroon" title="c16">fb</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E12:34_1_1_6"><i><font color="Green" title="E34">A46</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">fb</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E10:34_1_1_6"><i><font color="Green" title="E33">A45</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>
<br/>.= 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E21:34_1_1_6"><i><font color="Green" title="E39">A51</font></i></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></i>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #)
<b>by </b><i><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E13:34_1_1_6"><i><font color="Green" title="E35">A47</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="E23:34_1_1_6"/><b>then </b>
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E14:34_1_1_6"><i><font color="Green" title="E36">A48</font></i></a>, <a class="txt" href="#E20:34_1_1_6"><i><font color="Green" title="E38">A50</font></i></a>, <a class="ref" href="conlat_1.html#T31" title="CONLAT_1:th.31">CONLAT_1:31</a></i>;<br/>
<a NAME="E24:34_1_1_6"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a>  <a href="conlat_1.html#R2" title="CONLAT_1:pred.2">is-SubConcept-of</a> <font color="Maroon" title="c7">a'</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D19" title="CONLAT_1:def.19">CONLAT_1:def 19</a></i>;<br/>
<a NAME="E25:34_1_1_6"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">a'</font>
 
<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="E26:34_1_1_6"/><b>then </b>
<font color="Maroon" title="c8">b'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">a'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/>
<a NAME="E27:34_1_1_6"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E28:34_1_1_6"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p3">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>
<a NAME="E29:34_1_1_6"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#T20" title="LATTICE3:th.20">LATTICE3:20</a></i>;<br/>
<a NAME="E30:34_1_1_6"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E31:34_1_1_6"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:34_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K40</a>(<font color="Maroon" title="c4">f</font>) or  not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K40</a>(<font color="Maroon" title="c4">f</font>) or  not <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1_7"/><i><font color="Green" title="E28">A52</font></i>: 
( <font color="Maroon" title="c5">a</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">f</font> &amp; <font color="Maroon" title="c6">b</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">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c9">fa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:34_1_1_7"/><i><font color="Green" title="E29">A53</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">fa</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="c4">f</font>
 <b>by </b><i><a class="txt" href="#E1:34_1_1_7"><i><font color="Green" title="E28">A52</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="c10">O</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E6:34_1_1_7"/><i><font color="Green" title="E30">A54</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">fa</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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E4:34_1_1_7"><i><font color="Green" title="E29">A53</font></i></a></i>;<br/>
<i><font color="Green" title="E31">A55</font></i>: <font color="Maroon" title="c7">a</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E6:34_1_1_7"><i><font color="Green" title="E30">A54</font></i></a>, <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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</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/>
<b>consider </b><font color="Maroon" title="c12">fb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:34_1_1_7"/><i><font color="Green" title="E32">A56</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">fb</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="c4">f</font>
 <b>by </b><i><a class="txt" href="#E1:34_1_1_7"><i><font color="Green" title="E28">A52</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="c13">O'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E11:34_1_1_7"/><i><font color="Green" title="E33">A57</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">fb</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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">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="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E9:34_1_1_7"><i><font color="Green" title="E32">A56</font></i></a></i>;<br/>
<i><font color="Green" title="E34">A58</font></i>: <font color="Maroon" title="c8">b</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E11:34_1_1_7"><i><font color="Green" title="E33">A57</font></i></a>, <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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</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="E13:34_1_1_7"/><i><font color="Green" title="E35">A59</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fb</font>
 
<b>by </b><i><a class="txt" href="#E9:34_1_1_7"><i><font color="Green" title="E32">A56</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<i><font color="Green" title="E36">A60</font></i>: <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a>  = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c12">fb</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="#E11:34_1_1_7"><i><font color="Green" title="E33">A57</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">fb</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/>.= 
<font color="Maroon" title="c9">fa</font>
<b>by </b><i><a class="txt" href="#E1:34_1_1_7"><i><font color="Green" title="E28">A52</font></i></a>, <a class="txt" href="#E4:34_1_1_7"><i><font color="Green" title="E29">A53</font></i></a>, <a class="txt" href="#E13:34_1_1_7"><i><font color="Green" title="E35">A59</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">fa</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="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> 
<b>by </b><i><a class="txt" href="#E6:34_1_1_7"><i><font color="Green" title="E30">A54</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E37">A61</font></i>: the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) = 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #)
<b>by </b><i><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>
;<br/>
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c13">O'</font>,<font color="Maroon" title="c14">A'</font> #) = 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E14:34_1_1_7"><i><font color="Green" title="E36">A60</font></i></a>, <a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>
<br/>.= 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c10">O</font>,<font color="Maroon" title="c11">A</font> #)
<b>by </b><i><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E17:34_1_1_7"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E7:34_1_1_7"><i><font color="Green" title="E31">A55</font></i></a>, <a class="txt" href="#E12:34_1_1_7"><i><font color="Green" title="E34">A58</font></i></a>, <a class="txt" href="#E15:34_1_1_7"><i><font color="Green" title="E37">A61</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> , <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span> <b>by </b><i><a class="txt" href="#E11:34_1_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="#E13:34_1_1"><i><font color="Green" title="E26">A38</font></i></a>, <a class="txt" href="#E14:34_1_1"><i><font color="Green" title="E27">A39</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="E17:34_1_1"/>
for <font color="Olive" title="b1">CP</font> being <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">CP</font> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">CP</font> be  <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">CP</font> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a> </span><br/>

<a NAME="E1:34_1_1_8"/>
<font color="Maroon" title="c6">CP</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<a NAME="E2:34_1_1_8"/><b>then </b>
<font color="Maroon" title="c6">CP</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="c5">f</font>
 
<b>by </b><i><a class="txt" href="#E4:34_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <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="c7">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:34_1_1_8"/><i><font color="Green" title="E28">A62</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">CP</font>,<font color="Maroon" title="c7">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="c5">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/>
<b>consider </b><font color="Maroon" title="c8">Ov</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c9">Av</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E6:34_1_1_8"/><i><font color="Green" title="E29">A63</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">CP</font>,<font color="Maroon" title="c7">v</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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="#E4:34_1_1_8"><i><font color="Green" title="E28">A62</font></i></a></i>;<br/>
<a NAME="E7:34_1_1_8"/>
<font color="Maroon" title="c7">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="c5">f</font>
 
<b>by </b><i><a class="txt" href="#E4:34_1_1_8"><i><font color="Green" title="E28">A62</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="c10">v</font> = <font color="Maroon" title="c7">v</font> as  <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a>  <b>by </b><i><a class="txt" href="#E3:34_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></i>;<br/>
<i><font color="Green" title="E30">A64</font></i>: <font color="Maroon" title="c10">v</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E6:34_1_1_8"><i><font color="Green" title="E29">A63</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/>
<font color="Maroon" title="c6">CP</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</font> #) <a href="conlat_2.html#K8" title="CONLAT_2:func.8">.:</a> </span>)</span></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="#E6:34_1_1_8"><i><font color="Green" title="E29">A63</font></i></a>, <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#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c8">Ov</font>,<font color="Maroon" title="c9">Av</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/>
<b>hence </b><a NAME="E11:34_1_1_8"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">CP</font> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a> 
 <b>by </b><i><a class="txt" href="#E4:34_1_1_8"><i><font color="Green" title="E28">A62</font></i></a>, <a class="txt" href="#E9:34_1_1_8"><i><font color="Green" title="E30">A64</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:34_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> , <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span> st <br/>for <font color="Olive" title="b2">CP</font> being <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a> <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">CP</font> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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