
theorem Th31:
  for C being FormalContext for CP being object holds CP in B-carrier
  (C) iff CP is strict FormalConcept of C
proof
  let C be FormalContext;
  let CP be object;
A1: CP is strict FormalConcept of C implies CP in B-carrier(C)
  proof
    assume
A2: CP is strict FormalConcept of C;
    then reconsider CP as FormalConcept of C;
    set I9 = the Intent of CP;
    set E9 = the Extent of CP;
    (ObjectDerivation(C)).E9 = I9 & (AttributeDerivation(C)).I9 = E9 by Def9;
    hence thesis by A2;
  end;
  CP in B-carrier(C) implies CP is strict FormalConcept of C
  proof
    assume CP in B-carrier(C);
    then ex E being Subset of the carrier of C, I being Subset of the carrier'
    of C st CP = ConceptStr(#E,I#) & ConceptStr(#E,I#) is non empty & (
    ObjectDerivation(C)).E = I & (AttributeDerivation(C)).I = E;
    hence thesis by Def9;
  end;
  hence thesis by A1;
end;
