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

<b>set </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> :  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> </span>}</span> ;<br/>
<a NAME="E1:34_1_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a>  <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>,the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span>,the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice2.html#D2">LATTICE2:def 2</a></i>;<br/>
<a NAME="E2:34_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="conlat_1.html#K8">B-carrier</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10">B-join</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9">B-meet</a> <font color="Maroon">C</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="conlat_1.html#D23">CONLAT_1:def 23</a></i>;<br/>
<a NAME="E3:34_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="conlat_1.html#K11">ConceptLattice</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="conlat_1.html#K8">B-carrier</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10">B-join</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9">B-meet</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="conlat_1.html#D23">CONLAT_1:def 23</a></i>;<br/>
<a NAME="E4:34_1_1"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a>  <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="conlat_1.html#K8">B-carrier</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9">B-meet</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10">B-join</a> <font color="Maroon">C</font></span>)</span> #)
 
<b>by </b><i>, <a class="ref" href="lattice2.html#D2">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">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> :  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</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">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> :  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:34_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 ;<br/>
<a NAME="E4:34_1_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:34_1_1"><i><font color="Green">E23</font></i></a>, , <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<a NAME="E5:34_1_1_1"/>
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a>  is  <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:34_1_1_1"/><b>then </b>
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E1:34_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E7:34_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="txt" href="conlat_2.html#E3:34_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #),<span class="p3">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">O</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> :  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Olive">O</font>,<font color="Olive">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> </span>}</span>  as    <a href="relset_1.html#M2">Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E7:34_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">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">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>
 ;<br/>

<a NAME="E2:34_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</font> is  <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:34_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<b>then consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:34_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
 ;<br/>
<b>set </b><font color="Maroon">z</font> = <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> ;<br/>
<a NAME="E5:34_1_1_2"/>
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a>  is  <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> 
 
<b>by </b><i>, <a class="txt" href="conlat_2.html#E3:34_1_1"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="conlat_2.html#E1:34_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<a NAME="E7:34_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="conlat_2.html#E3:34_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:34_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:34_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E9:34_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_3"/><i><font color="Green">E46</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:34_1_1_3"/><i><font color="Green">E47</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 ;<br/>
<b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:34_1_1_3"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:34_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:34_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><font color="Maroon">y1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:34_1_1"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:34_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>  st <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">b</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E1:34_1_1_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E3:34_1_1_4"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">fa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:34_1_1_4"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E7:34_1_1_4"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>
<i><font color="Green">E73</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E9:34_1_1_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fa</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon">fb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:34_1_1_4"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">fb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E13:34_1_1_4"/><i><font color="Green">E77</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">fb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E79</font></i>: <font color="Maroon">b</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E15:34_1_1_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fb</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E16:34_1_1_4"/>
<font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:34_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E17:34_1_1_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K4">%</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">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">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K4">%</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="relset_1.html#K6">~</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T20">LATTICE3:20</a></i>;<br/>
<a NAME="E19:34_1_1_4"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K4">%</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font> <a href="lattice3.html#K4">%</a> , <font color="Maroon">bb</font> = <font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E21:34_1_1_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">bb</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">aa</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E22:34_1_1_4"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">aa</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">aa</font>
 
;<br/>
<a NAME="E23:34_1_1_4"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">bb</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font>
 
;<br/>
<a NAME="E24:34_1_1_4"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">bb</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">aa</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E25:34_1_1_4"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">bb</font></span>)</span> <a href="conlat_1.html#K12">@</a>  <a href="conlat_1.html#R2">is-SubConcept-of</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">aa</font></span>)</span> <a href="conlat_1.html#K12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#T47">CONLAT_1:47</a></i>;<br/>
<a NAME="E26:34_1_1_4"/>
<font color="Maroon">fa</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fa</font> = <font color="Maroon">fa</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E28:34_1_1_4"/>
<font color="Maroon">fb</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fb</font> = <font color="Maroon">fb</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> ;<br/>
<font color="Maroon">fa</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>then </b><i><font color="Green">E118</font></i>: the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">fa</font> <a href="conlat_1.html#K12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
<b>by </b><i/>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
;<br/>
<font color="Maroon">fb</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>then </b>the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">fb</font> <a href="conlat_1.html#K12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
<b>by </b><i/>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
;<br/>
<a NAME="E34:34_1_1_4"/><b>then </b>
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">fb</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">fa</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="conlat_1.html#D19">CONLAT_1:def 19</a></i>;<br/>
<a NAME="E35:34_1_1_4"/><b>then </b>
<font color="Maroon">fa</font> <a href="conlat_1.html#K12">@</a>  <a href="conlat_1.html#R2">is-SubConcept-of</a> <font color="Maroon">fb</font> <a href="conlat_1.html#K12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#T31">CONLAT_1:31</a></i>;<br/>
<b>hence </b><a NAME="E36:34_1_1_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="conlat_1.html#T47">CONLAT_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:34_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">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">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1_1_5"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">O'</font> = the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="conlat_1.html#K12">@</a> </span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> ;<br/>
<b>reconsider </b><font color="Maroon">A'</font> = the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="conlat_1.html#K12">@</a> </span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font> ;<br/>
<b>set </b><font color="Maroon">CP</font> =  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #);<br/>
<i><font color="Green">E119</font></i>: <span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #) = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #)
<b>by </b><i><a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>
;<br/>
<i><font color="Green">E120</font></i>: <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #) = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #)
<b>by </b><i><a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>
;<br/>
<a NAME="E7:34_1_1_5"/>
(  not the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="conlat_1.html#K12">@</a> </span>)</span> is <a href="xboole_0.html#V1">empty</a> or  not the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="conlat_1.html#K12">@</a> </span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i><a class="ref" href="conlat_1.html#D11">CONLAT_1:def 11</a></i>;<br/>
<a NAME="E8:34_1_1_5"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E4:34_1_1"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="conlat_1.html#D11">CONLAT_1:def 11</a>, <a class="ref" href="conlat_1.html#D13">CONLAT_1:def 13</a></i>;<br/>
<a NAME="E9:34_1_1_5"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:34_1_1_5"/><b>then </b>
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #) <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:34_1_1_5"/><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #),<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Ov</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">Av</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E14:34_1_1_5"/><i><font color="Green">E124</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #),<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E15:34_1_1_5"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as  <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a>  <b>by </b><i><a class="txt" href="conlat_2.html#E1:34_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<i><font color="Green">E128</font></i>: <font color="Maroon">v</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<i><font color="Green">E130</font></i>:  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #) = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #)
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E19:34_1_1_5"/><b>then </b><i><font color="Green">E131</font></i>: 
the <a href="conlat_1.html#U4">Extent</a> of <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:34_1_1_5"/>
the <a href="conlat_1.html#U5">Intent</a> of <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E21:34_1_1_5"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:34_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E14:34_1_1"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E1:34_1_1_6"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E3:34_1_1_6"/><i><font color="Green">E135</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">fa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:34_1_1_6"/><i><font color="Green">E138</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E7:34_1_1_6"/><i><font color="Green">E140</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E141</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>consider </b><font color="Maroon">fb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:34_1_1_6"/><i><font color="Green">E143</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">fb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E12:34_1_1_6"/><i><font color="Green">E144</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">fb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E146</font></i>: <font color="Maroon">b</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E14:34_1_1_6"/><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a>  <a href="conlat_1.html#R2">is-SubConcept-of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="conlat_1.html#K12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="conlat_1.html#T47">CONLAT_1:47</a></i>;<br/>
<a NAME="E15:34_1_1_6"/>
<font color="Maroon">fa</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fa</font> = <font color="Maroon">fa</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E17:34_1_1_6"/>
<font color="Maroon">fb</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fb</font> = <font color="Maroon">fb</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> ;<br/>
<i><font color="Green">E148</font></i>: <font color="Maroon">fa</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<i><font color="Green">E149</font></i>: the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">fa</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
<br/>.= 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
<b>by </b><i/>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
;<br/>
<i><font color="Green">E150</font></i>: <font color="Maroon">fb</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">fb</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
<br/>.= 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
<b>by </b><i/>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
;<br/>
<a NAME="E23:34_1_1_6"/><b>then </b>
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="conlat_1.html#T31">CONLAT_1:31</a></i>;<br/>
<a NAME="E24:34_1_1_6"/><b>then </b>
<font color="Maroon">b'</font> <a href="conlat_1.html#K12">@</a>  <a href="conlat_1.html#R2">is-SubConcept-of</a> <font color="Maroon">a'</font> <a href="conlat_1.html#K12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D19">CONLAT_1:def 19</a></i>;<br/>
<a NAME="E25:34_1_1_6"/><b>then </b>
<font color="Maroon">b'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a'</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#T47">CONLAT_1:47</a></i>;<br/>
<a NAME="E26:34_1_1_6"/><b>then </b>
<font color="Maroon">b'</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a'</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E27:34_1_1_6"/><b>then </b><i><font color="Green">E151</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="lattice3.html#K4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="lattice3.html#K4">%</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E28:34_1_1_6"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K4">%</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p3">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D7">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">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K4">%</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K4">%</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#T20">LATTICE3:20</a></i>;<br/>
<a NAME="E30:34_1_1_6"/><b>then </b>
<font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E31:34_1_1_6"/>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:34_1_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">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">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:34_1_1_7"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> </span>)</span> ;<br/>
<b>consider </b><font color="Maroon">fa</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:34_1_1_7"/><i><font color="Green">E153</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E6:34_1_1_7"/><i><font color="Green">E154</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a></i>;<br/>
<i><font color="Green">E155</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>consider </b><font color="Maroon">fb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:34_1_1_7"/><i><font color="Green">E157</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">fb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E11:34_1_1_7"/><i><font color="Green">E158</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">fb</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E159</font></i>: <font color="Maroon">b</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #)
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E13:34_1_1_7"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fb</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<i><font color="Green">E161</font></i>: <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) <a href="conlat_2.html#K8" target="_self">.:</a>  = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">fb</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">fb</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">fa</font>
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">fa</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>then </b><i><font color="Green">E162</font></i>: the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) = 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="conlat_1.html#U4">Extent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
<b>by </b><i/>
;<br/>
the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) = 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="conlat_1.html#U5">Intent</a> of <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #)
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E17:34_1_1_7"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="lattice4.html#M1">Homomorphism</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> , <a href="conlat_1.html#K11">ConceptLattice</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> <b>by </b><i><a class="txt" href="conlat_2.html#E3:34_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="conlat_2.html#E4:34_1_1"><i><font color="Green">E26</font></i></a>, , </i>;<br/>
<a NAME="E17:34_1_1"/>
for <font color="Olive">CP</font> being <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">CP</font> <a href="hidden.html#R1">=</a> <font color="Olive">CP</font> <a href="conlat_2.html#K9" target="_self">.:</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">CP</font> be  <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:34_1_1_8"/>
<font color="Maroon">CP</font> <a href="hidden.html#R2">in</a>  <a href="conlat_1.html#K8">B-carrier</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<a NAME="E2:34_1_1_8"/><b>then </b>
<font color="Maroon">CP</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="conlat_2.html#E2:34_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:34_1_1_8"/><i><font color="Green">E163</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">CP</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Ov</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">Av</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E6:34_1_1_8"/><i><font color="Green">E164</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">CP</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) is   <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:34_1_1_8"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as  <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a>  <b>by </b><i><a class="txt" href="conlat_2.html#E1:34_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<i><font color="Green">E165</font></i>: <font color="Maroon">v</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<font color="Maroon">CP</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #),<span class="p2">(<span class="default"><a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #) <a href="conlat_2.html#K8" target="_self">.:</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">Ov</font>,<font color="Maroon">Av</font> #)
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E11:34_1_1_8"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">CP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CP</font> <a href="conlat_2.html#K9" target="_self">.:</a> 
 <b>by </b><i><a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<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">Homomorphism</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> <a href="lattice2.html#K1">.:</a> , <a href="conlat_1.html#K11">ConceptLattice</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> st <br/>for <font color="Olive">CP</font> being <a href="conlat_1.html#V6">strict</a> <a href="conlat_1.html#NM4">FormalConcept</a> of <font color="Maroon">C</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">CP</font> <a href="hidden.html#R1">=</a> <font color="Olive">CP</font> <a href="conlat_2.html#K9" target="_self">.:</a> 
 ;<br/>


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