
theorem :: WWACorA: :: Bill: Is this the right translation
  for X being finite non empty set, F, G being Dependency-set of X st
charact_set F = charact_set G holds Dependency-closure F = Dependency-closure G
proof
  let A be finite non empty set, F, G be Dependency-set of A such that
A1: charact_set F = charact_set G;
  set DCF = Dependency-closure F, DCG = Dependency-closure G;
  now
    let x be object;
    hereby
      assume
A2:   x in DCF;
      then consider a, b being Subset of A such that
A3:   x = [a,b] by Th9;
      for c be Subset of A st a c= c & not b c= c holds c in charact_set G
      by A1,A2,A3,Th57;
      hence x in DCG by A3,Th57;
    end;
    assume
A4: x in DCG;
    then consider a, b being Subset of A such that
A5: x = [a,b] by Th9;
    for c be Subset of A st a c= c & not b c= c holds c in charact_set F
    by A1,A4,A5,Th57;
    hence x in DCF by A5,Th57;
  end;
  hence thesis by TARSKI:2;
end;
