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<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>reconsider </b><font color="Maroon">g</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="conlat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" target="_self">Context</a> <font color="Maroon">L</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">d</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="conlat_1.html#U2">Attributes</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" target="_self">Context</a> <font color="Maroon">L</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:24"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="lattice6.html#V8">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice6.html#D13">LATTICE6:def 13</a><br/>

<a NAME="E1:24_1"/><i><font color="Green">E23</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T43">LATTICE3:43</a></i>;<br/>
<a NAME="E2:24_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">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">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:24"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">D'</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font></span>)</span> st <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">D'</font>,<font color="Maroon">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E1:24_2"/><i><font color="Green">E25</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T43">LATTICE3:43</a></i>;<br/>
<a NAME="E2:24_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:24_2"/>
 ex <font color="Olive">D'</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font></span>)</span> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">D'</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:24"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">d</font> is <a href="lattice6.html#V9">infimum-dense</a>
 
<b>by </b><i><a class="ref" href="lattice6.html#D14">LATTICE6:def 14</a></i>;<br/>
<a NAME="E6:24"/>
for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" target="_self">Context</a> <font color="Maroon">L</font></span>)</span><br/> for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" target="_self">Context</a> <font color="Maroon">L</font></span>)</span> holds <br/> ( <font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> iff <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">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">o</font> be   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" target="_self">Context</a> <font color="Maroon">L</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" target="_self">Context</a> <font color="Maroon">L</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">o</font>, <font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E2:24_3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font> iff <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U3">Information</a> of <span class="p1">(<span class="default"><a href="conlat_2.html#K6" target="_self">Context</a> <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="conlat_1.html#D5">CONLAT_1:def 5</a></i>;<br/>
<a NAME="E3:24_3"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> &amp; <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<b>hence </b><a NAME="E4:24_3"/>
( <font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font> iff <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:24"/>
 <a href="conlat_1.html#K11">ConceptLattice</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K6" target="_self">Context</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">L</font> <a href="filter_1.html#R1">are_isomorphic</a> 
 <b>by </b><i>, , </i>;<br/>


</div>
