
theorem
  for C being FormalContext for CP being strict FormalConcept of C holds
  (B-meet(C)).(CP,Concept-with-all-Objects(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-meet(C)).(CP,Concept-with-all-Objects(C)) = ConceptStr(#O,A#) and
A2: O = (the Extent of CP) /\ (the Extent of Concept-with-all-Objects(C) ) and
A3: A = (ObjectDerivation(C)).((AttributeDerivation(C)). ((the Intent of
  CP) \/ (the Intent of Concept-with-all-Objects(C)))) by Def17;
A4: O = (the Extent of CP) /\ the carrier of C by A2,Th23
    .= the Extent of CP by XBOOLE_1:28;
  the carrier of C c= the carrier of C;
  then reconsider O9 = the carrier of C as Subset of the carrier of C;
A5: (ObjectDerivation(C)).(the Extent of CP) \/ (ObjectDerivation(C)).O9 = (
  ObjectDerivation(C)).(the Extent of CP) by Th3,XBOOLE_1:12;
  A = (ObjectDerivation(C)).((AttributeDerivation(C)). ((the Intent of CP)
  \/ (ObjectDerivation(C)).(the Extent of Concept-with-all-Objects(C)))) by A3
,Def9
    .= (ObjectDerivation(C)).((AttributeDerivation(C)). ((the Intent of CP)
  \/ (ObjectDerivation(C)). the carrier of C)) by Th23
    .= (ObjectDerivation(C)).((AttributeDerivation(C)). ((ObjectDerivation(C
  )).(the Extent of CP))) by A5,Def9
    .= (ObjectDerivation(C)).(the Extent of CP) by Th7
    .= the Intent of CP by Def9;
  hence thesis by A1,A4;
end;
