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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> </span><br/>

<b>reconsider </b><font color="Maroon" title="c2">g</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">d</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:24"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font> is <a href="lattice6.html#V8" title="LATTICE6:attr.8">supremum-dense</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice6.html#D13" title="LATTICE6:def.13">LATTICE6:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font></span>)</span> st <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E1:24_1"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T43" title="LATTICE3:th.43">LATTICE3:43</a></i>;<br/>
<a NAME="E2:24_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">g</font></span>)</span> st <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E1:24_1"><i><font color="Green" title="E17">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:24"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">D'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">d</font></span>)</span> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Olive" title="b2">D'</font>,<font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">D'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">d</font></span>)</span> st <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Olive" title="b1">D'</font>,<font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E1:24_2"/><i><font color="Green" title="E17">A3</font></i>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T43" title="LATTICE3:th.43">LATTICE3:43</a></i>;<br/>
<a NAME="E2:24_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:24_2"/>
 ex <font color="Olive" title="b1">D'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">d</font></span>)</span> st <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Olive" title="b1">D'</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="conlat_2.html#E1:24_2"><i><font color="Green" title="E17">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:24"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">d</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a>
 
<b>by </b><i><a class="ref" href="lattice6.html#D14" title="LATTICE6:def.14">LATTICE6:def 14</a></i>;<br/>
<a NAME="E6:24"/>
for <font color="Olive" title="b1">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span><br/> for <font color="Olive" title="b2">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> iff <font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">o</font> be   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span> holds <br/> ( <font color="Maroon" title="c4">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font> iff <font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> )</span><br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c5">a</font> iff <font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c6">o'</font> = <font color="Maroon" title="c4">o</font>, <font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E2:24_3"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c4">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c5">a</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U3" title="CONLAT_1:sel.3">Information</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="conlat_1.html#D5" title="CONLAT_1:def.5">CONLAT_1:def 5</a></i>;<br/>
<a NAME="E3:24_3"/>
( <font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o'</font> &amp; <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a'</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>;<br/>
<b>hence </b><a NAME="E4:24_3"/>
( <font color="Maroon" title="c4">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c5">a</font> iff <font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">o</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="conlat_2.html#E2:24_3"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:24"/>
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="conlat_2.html#E3:24"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="conlat_2.html#E5:24"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="conlat_2.html#T14" target="_self" title="CONLAT_2:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
