
theorem Th36: ::  WWA3c:
  for X being finite non empty set, F being Dependency-set of X
  holds enclosure_of F is (B1) (B2)
proof
  let X be finite non empty set, F be Dependency-set of X;
  set B = enclosure_of F;
A1: for x, y being Subset of X st [x, y] in F & x c= X holds y c= X;
  X = [#]X;
  then X in B by A1;
  hence B is (B1);
  let a, b be set such that
A2: a in B and
A3: b in B;
  consider b9 being Subset of X such that
A4: b9 = b and
A5: for x, y being Subset of X st [x, y] in F & x c= b9 holds y c= b9 by A3;
  consider a9 being Subset of X such that
A6: a9 = a and
A7: for x, y being Subset of X st [x, y] in F & x c= a9 holds y c= a9 by A2;
  reconsider ab = a9 /\ b9 as Subset of X;
  now
    let x, y be Subset of X such that
A8: [x, y] in F and
A9: x c= ab;
A10: y c= b9 by A5,A8,A9,XBOOLE_1:18;
    y c= a9 by A7,A8,A9,XBOOLE_1:18;
    hence y c= ab by A10,XBOOLE_1:19;
  end;
  hence thesis by A6,A4;
end;
