
theorem
  for C being FormalContext for O being Subset of the carrier of C holds
  (ex A being Subset of the carrier' of C st ConceptStr(#O,A#) is FormalConcept
  of C) iff (AttributeDerivation(C)).((ObjectDerivation(C)).O) = O
proof
  let C be FormalContext;
  let O be Subset of the carrier of C;
A1: now
    O c= (AttributeDerivation(C)).((ObjectDerivation(C)).O) by Th5;
    then
A2: for x being object holds x in O implies x in (AttributeDerivation(C)).((
    ObjectDerivation(C)).O);
    given A being Subset of the carrier' of C such that
A3: ConceptStr(#O,A#) is FormalConcept of C;
    (AttributeDerivation(C)).((ObjectDerivation(C)).O) c= O by A3,Th19;
    then
    for x being object holds x in (AttributeDerivation(C)).((ObjectDerivation
    (C)).O) implies x in O;
    hence (AttributeDerivation(C)).((ObjectDerivation(C)).O) = O by A2,TARSKI:2
;
  end;
  now
    reconsider A = (ObjectDerivation(C)).O as Subset of the carrier' of C;
    set M9 = ConceptStr(#O,A#);
    assume (AttributeDerivation(C)).((ObjectDerivation(C)).O) = O;
    then M9 is FormalConcept of C by Def9,Lm1;
    hence ex A being Subset of the carrier' of C st ConceptStr(#O,A#) is
    FormalConcept of C;
  end;
  hence thesis by A1;
end;
