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

<b>let </b><font color="Maroon">C</font> be   <a href="conlat_1.html#NM1">FormalContext</a>;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font>;<br/>
<a NAME="E1:35"/>
 <a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><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="E1:35_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <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/>
<a NAME="E3:35_1"/>
<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/>
<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"><a href="conlat_1.html#K11">ConceptLattice</a> <font color="Maroon">C</font></span>)</span> ;<br/>
<a NAME="E5:35_1"/><i><font color="Green">E23</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="conlat_2.html#K9" target="_self">.:</a>  &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="conlat_2.html#K9" target="_self">.:</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E6:35_1"/><i><font color="Green">E25</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/>
<b>then </b><i><font color="Green">E26</font></i>: the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="conlat_1.html#K12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="conlat_2.html#K9" target="_self">.:</a> </span>)</span>
<b>by </b><i><a class="txt" href="conlat_2.html#E1:35_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="conlat_2.html#E5:35_1"><i><font color="Green">E23</font></i></a>, </i>
<br/>.= 
the <a href="conlat_1.html#U5">Intent</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/>
;<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> = 
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">b</font> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="conlat_2.html#K9" target="_self">.:</a> </span>)</span>
<b>by </b><i><a class="txt" href="conlat_2.html#E1:35_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="conlat_2.html#E5:35_1"><i><font color="Green">E23</font></i></a>, , </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/>
;<br/>
<b>then </b><font color="Maroon">a</font> = 
<font color="Maroon">b</font> <a href="conlat_1.html#K12">@</a> 
<b>by </b><i><a class="txt" href="conlat_2.html#E6:35_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>
;<br/>
<b>hence </b><a NAME="E10:35_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font> is <a href="lattice4.html#V1">monomorphism</a>
 
<b>by </b><i><a class="ref" href="lattice4.html#D2">LATTICE4:def 2</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green">E40</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="E4:35"/><i><font color="Green">E41</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="E5:35"/>
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> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><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="E1:35_2"/>
<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">E43</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">E44</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="E7:35_2"/><i><font color="Green">E46</font></i>: 
(  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="E8:35_2"/><b>then </b>
 <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="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="E9:35_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <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> 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>
 
<b>by </b><i>, <a class="ref" href="conlat_1.html#T35">CONLAT_1:35</a></i>;<br/>
<b>reconsider </b><font color="Maroon">CP</font> =  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">A'</font>,<font color="Maroon">O'</font> #) as  <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#E2:35"><i><font color="Green">E27</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="E11:35_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">CP</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="conlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:35_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <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/>;<br/>
<a NAME="E13:35_2"/><i><font color="Green">E51</font></i>: 
the <a href="conlat_1.html#U5">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">CP</font> <a href="conlat_2.html#K9" target="_self">.:</a> </span>)</span> <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/>;<br/>
<a NAME="E14:35_2"/>
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">CP</font> <a href="conlat_2.html#K9" target="_self">.:</a> </span>)</span> <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/>;<br/>
<a NAME="E15:35_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">CP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 
<b>by </b><i>, , <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/>
<b>hence </b><a NAME="E16:35_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="conlat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:35"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font></span>)</span> <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="E7:35"/><b>then </b>
 <a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font> is <a href="lattice4.html#V2">epimorphism</a>
 
<b>by </b><i><a class="ref" href="lattice4.html#D3">LATTICE4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:35"/>
 <a href="conlat_2.html#K10" target="_self">DualHomomorphism</a> <font color="Maroon">C</font> is <a href="lattice4.html#V3">isomorphism</a>
 <b>by </b><i>, <a class="ref" href="lattice4.html#D4">LATTICE4:def 4</a></i>;<br/>


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