
theorem Th4:
  for C being FormalContext for A1,A2 being Subset of the carrier'
of C holds A1 c= A2 implies (AttributeDerivation(C)).A2 c= (AttributeDerivation
  (C)).A1
proof
  let C be FormalContext;
  let A1,A2 be Subset of the carrier' of C;
  assume
A1: A1 c= A2;
  let x be object;
  assume x in (AttributeDerivation(C)).A2;
  then x in {o where o is Object of C : for a being Attribute of C st a in A2
  holds o is-connected-with a} by Def3;
  then consider x9 being Object of C such that
A2: x9 = x and
A3: for a being Attribute of C st a in A2 holds x9 is-connected-with a;
  for a being Attribute of C st a in A1 holds x9 is-connected-with a by A1,A3;
  then x in {o where o is Object of C : for a being Attribute of C st a in A1
  holds o is-connected-with a} by A2;
  hence thesis by Def3;
end;
