
theorem
  for C being FormalContext for CP being strict FormalConcept of C holds
  (B-join(C)).(CP,Concept-with-all-Attributes(C)) = CP
proof
  let C be FormalContext;
  let CP be strict FormalConcept of C;
  consider O being Subset of the carrier of C, A being Subset of the carrier'
  of C such that
A1: (B-join(C)).(CP,Concept-with-all-Attributes(C)) = ConceptStr(#O,A#) and
A2: O = (AttributeDerivation(C)).((ObjectDerivation(C)). ((the Extent of
  CP) \/ (the Extent of Concept-with-all-Attributes(C)))) and
A3: A = (the Intent of CP) /\ (the Intent of Concept-with-all-Attributes
  (C)) by Def18;
A4: A = (the Intent of CP) /\ the carrier' of C by A3,Th23
    .= the Intent of CP by XBOOLE_1:28;
  the carrier' of C c= the carrier' of C;
  then reconsider A9 = the carrier' of C as Subset of the carrier' of C;
A5: (AttributeDerivation(C)).(the Intent of CP) \/ (AttributeDerivation(C) )
  .A9 = (AttributeDerivation(C)).(the Intent of CP) by Th4,XBOOLE_1:12;
  O = (AttributeDerivation(C)).((ObjectDerivation(C)). ((the Extent of CP)
\/ (AttributeDerivation(C)).(the Intent of Concept-with-all-Attributes(C))))
by A2,Def9
    .= (AttributeDerivation(C)).((ObjectDerivation(C)). ((the Extent of CP)
  \/ (AttributeDerivation(C)). the carrier' of C)) by Th23
    .= (AttributeDerivation(C)).((ObjectDerivation(C)). ((
  AttributeDerivation(C)).(the Intent of CP))) by A5,Def9
    .= (AttributeDerivation(C)).(the Intent of CP) by Th8
    .= the Extent of CP by Def9;
  hence thesis by A1,A4;
end;
