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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">f</font> =  <a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font>;<br/>
<a NAME="E1:35"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="conlat_1.html#K8" title="CONLAT_1:func.8">B-carrier</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K10" title="CONLAT_1:func.10">B-join</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="conlat_1.html#K9" title="CONLAT_1:func.9">B-meet</a> <font color="Maroon" title="c1">C</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D23" title="CONLAT_1:def.23">CONLAT_1:def 23</a></span>;<br/>
<a NAME="E2:35"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#D2" title="LATTICE2:def.2">LATTICE2:def 2</a></span>;<br/>
<a NAME="E3:35"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> or <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">u</font> = <font color="Maroon" title="c3">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">A9</font> = the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">O9</font> = the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">CP</font> =  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">O9</font> #);<br/>
<a NAME="E5:35_1"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
(  not the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D11" title="CONLAT_1:def.11">CONLAT_1:def 11</a></span>;<br/>
<span class="lab"><font color="Green" title="E25">A4</font></span>: <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">O9</font> #) = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_2.html#T19" target="_self" title="CONLAT_2:th.19">Th19</a></span>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">O9</font> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></span>
;<br/>
<span class="lab"><font color="Green" title="E26">A5</font></span>: <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">O9</font> #) = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_2.html#T18" target="_self" title="CONLAT_2:th.18">Th18</a></span>
<br/>.= 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">O9</font> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></span>
;<br/>
<a NAME="E8:35_1"/><span class="kw">then </span>
 <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">O9</font> #) is   <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="conlat_1.html#D11" title="CONLAT_1:def.11">CONLAT_1:def 11</a>, <a class="ref" href="conlat_1.html#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></span>;<br/>
<a NAME="E9:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A6</font></span>: 
 <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">O9</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="conlat_1.html#T35" title="CONLAT_1:th.35">CONLAT_1:35</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">CP</font> =  <a href="conlat_1.html#G2" title="CONLAT_1:aggr.2">ConceptStr</a>(# <font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">O9</font> #) as   <a href="conlat_1.html#V4" title="CONLAT_1:attr.4">strict</a>  <a href="conlat_1.html#NM4" title="CONLAT_1:NM.4">FormalConcept</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E6:35_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="conlat_1.html#D11" title="CONLAT_1:def.11">CONLAT_1:def 11</a>, <a class="ref" href="conlat_1.html#D13" title="CONLAT_1:def.13">CONLAT_1:def 13</a></span>;<br/>
<a NAME="E11:35_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">CP</font> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></span>;<br/>
<a NAME="E12:35_1"/>
( <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">CP</font> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a>  &amp; the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">CP</font> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a>, <a class="ref" href="conlat_2.html#D9" target="_self" title="CONLAT_2:def.9">Def9</a></span>;<br/>
<a NAME="E13:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">CP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></span>;<br/>
<a NAME="E14:35_1"/>
<font color="Maroon" title="c8">CP</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:35_1"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:35"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="conlat_2.html#K7" title="CONLAT_2:func.7">.:</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E5:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>: 
 <a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E6:35"/>
 <a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K41</a>(<span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span>) or  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K41</a>(<span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span>) or  not <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_2"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E2:35_2"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/>
<a NAME="E4:35_2"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#D2" title="LATTICE2:def.2">LATTICE2:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<a NAME="E6:35_2"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
( <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></span>;<br/>
<a NAME="E7:35_2"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
( <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a>  &amp; <span class="p1">(<span class="default"><a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_2.html#D9" target="_self" title="CONLAT_2:def.9">Def9</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E29">A14</font></span>: the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E6:35_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></span>
<br/>.= 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></span>
;<br/>
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> = 
the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span> <a href="conlat_2.html#K9" title="CONLAT_2:func.9">.:</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E7:35_2"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E6:35_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></span>
<br/>.= 
the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_2.html#D8" target="_self" title="CONLAT_2:def.8">Def8</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c7">a</font> = 
<font color="Maroon" title="c8">b</font> <a href="conlat_1.html#K12" title="CONLAT_1:func.12">@</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#D24" title="CONLAT_1:def.24">CONLAT_1:def 24</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:35_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:35"/>
 <a href="conlat_2.html#K10" title="CONLAT_2:func.10">DualHomomorphism</a> <font color="Maroon" title="c1">C</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E24">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
