
theorem Th3:
  for C being FormalContext for O1,O2 being Subset of the carrier
of C holds O1 c= O2 implies (ObjectDerivation(C)).O2 c= (ObjectDerivation(C)).
  O1
proof
  let C be FormalContext;
  let O1,O2 be Subset of the carrier of C;
  assume
A1: O1 c= O2;
  let x be object;
  assume x in (ObjectDerivation(C)).O2;
  then x in {a where a is Attribute of C : for o being Object of C st o in O2
  holds o is-connected-with a} by Def2;
  then consider x9 being Attribute of C such that
A2: x9 = x and
A3: for o being Object of C st o in O2 holds o is-connected-with x9;
  for o being Object of C st o in O1 holds o is-connected-with x9 by A1,A3;
  then x in {a where a is Attribute of C : for o being Object of C st o in O1
  holds o is-connected-with a} by A2;
  hence thesis by Def2;
end;
