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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="conlat_1.html#NM1">FormalContext</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" target="_self">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="conlat_1.html#K11" target="_self">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:90_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/><a NAME="E3:90_1"/><b>then </b>
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" target="_self">ConceptLattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>;<br/><a NAME="E4:90_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" target="_self">B-join</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> 
 ;<br/><a NAME="E5:90_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" target="_self">B-join</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> 
 ;<br/><a NAME="E6:90_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" target="_self">B-join</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:90_1"/><i><font color="Green">E52</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_1.html#K10" target="_self">B-join</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> #) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="subset_1.html#K4">\/</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="subset_1.html#K9">/\</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="conlat_1.html#D22" target="_self">Def22</a></i>;<br/><a NAME="E9:90_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>
 <b>by </b><i><a class="txt" href="conlat_1.html#E6:90_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="conlat_1.html#E8:90_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:90_1"/><b>then </b>
the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E11:90_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a>  <a href="conlat_1.html#R2" target="_self">is-SubConcept-of</a> <font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> 
 <b>by </b><i><a class="ref" href="conlat_1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a>  <a href="conlat_1.html#R2" target="_self">is-SubConcept-of</a> <font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> 
 ;<br/><a NAME="E2:90_2"/><b>then </b>
the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="conlat_1.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E3:90_2"/><b>then </b><i><font color="Green">E52</font></i>: 
the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="subset_1.html#K9">/\</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E4:90_2"/>
the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:90_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="conlat_1.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E5:90_2"/><b>then </b><i><font color="Green">E53</font></i>: 
the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="subset_1.html#K4">\/</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:90_2"/><i><font color="Green">E54</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_1.html#K10" target="_self">B-join</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3" target="_self">ConceptStr</a>(# <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> #) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="subset_1.html#K4">\/</a> the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="subset_1.html#K9">/\</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="conlat_1.html#D22" target="_self">Def22</a></i>;<br/><font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> the <a href="conlat_1.html#U5" target="_self">Intent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>
<b>by </b><i><a class="txt" href="conlat_1.html#E5:90_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="conlat_1.html#E7:90_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="conlat_1.html#D13" target="_self">Def13</a></i>
<br/>.= 
the <a href="conlat_1.html#U4" target="_self">Extent</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="conlat_1.html#D13" target="_self">Def13</a></i>
;<br/><a NAME="E9:90_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" target="_self">B-join</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E3:90_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="conlat_1.html#E7:90_2"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E10:90_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" target="_self">B-join</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E11:90_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_1.html#K10" target="_self">B-join</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E12:90_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>;<br/><b>hence </b><a NAME="E13:90_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:90"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="conlat_1.html#K12" target="_self">@</a>  <a href="conlat_1.html#R2" target="_self">is-SubConcept-of</a> <font color="Maroon">c<sub>3</sub></font> <a href="conlat_1.html#K12" target="_self">@</a>  )
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:90"><i><font color="Green">E50</font></i></a></i>;<br/>


</div>
