
theorem
  for C being FormalContext for A being Subset of the carrier' of C
  holds (ex O being Subset of the carrier of C st ConceptStr(#O,A#) is
FormalConcept of C) iff (ObjectDerivation(C)).((AttributeDerivation(C)).A) = A
proof
  let C be FormalContext;
  let A be Subset of the carrier' of C;
A1: now
    A c= (ObjectDerivation(C)).((AttributeDerivation(C)).A) by Th6;
    then
A2: for x being object holds x in A implies x in (ObjectDerivation(C)).((
    AttributeDerivation(C)).A);
    given O being Subset of the carrier of C such that
A3: ConceptStr(#O,A#) is FormalConcept of C;
    (ObjectDerivation(C)).((AttributeDerivation(C)).A) c= A by A3,Th21;
    then
    for x being object holds x in (ObjectDerivation(C)).((AttributeDerivation
    (C)).A) implies x in A;
    hence (ObjectDerivation(C)).((AttributeDerivation(C)).A) = A by A2,TARSKI:2
;
  end;
  now
    reconsider O = (AttributeDerivation(C)).A as Subset of the carrier of C;
    set M9 = ConceptStr(#O,A#);
    assume (ObjectDerivation(C)).((AttributeDerivation(C)).A) = A;
    then M9 is FormalConcept of C by Def9,Lm1;
    hence ex O being Subset of the carrier of C st ConceptStr(#O,A#) is
    FormalConcept of C;
  end;
  hence thesis by A1;
end;
