
theorem Th38: ::  WWA3d:
  for X being finite non empty set, F being Dependency-set of X
  holds Dependency-closure F = X deps_encl_by enclosure_of F
proof
  let X be finite non empty set, F be Dependency-set of X;
  set B = enclosure_of F;
  set H = X deps_encl_by B;
  reconsider H as Full-family of X by Th33;
A1: for G being Dependency-set of X st F c= G & G is full_family holds H c=
  G by Th37;
  F c= H by Th37;
  hence thesis by A1,Def24;
end;
