
theorem :: APP1:
  for X being set, F being Dependency-set of X holds F is (F1) (F2) (F3)
  (F4) iff F is (DC1) (DC3) (DC4)
proof
  let X be set, F be Dependency-set of X;
  hereby
    assume that
A1: F is (F1) and
A2: F is (F2) and
A3: F is (F3) and
A4: F is (F4);
    thus F is (DC1) by A2;
    thus F is (DC3) by A1,A3;
    thus F is (DC4)
    proof
      let A, B, C be Subset of X;
      assume that
A5:   [A, B] in F and
A6:   [A, C] in F;
      [A\/A, B\/C] in F by A4,A5,A6;
      hence thesis;
    end;
  end;
  assume that
A7: F is (DC1) and
A8: F is (DC3) and
A9: F is (DC4);
  thus F is (F1) by A8;
  thus F is (F2) by A7;
  thus F is (F3) by A7,A8;
  let A, B, A9, B9 be Subset of X such that
A10: [A, B] in F and
A11: [A9, B9] in F;
  A9 c= A\/A9 by XBOOLE_1:7;
  then [A\/A9, A9] in F by A8;
  then
A12: [A\/A9, B9] in F by A7,A11,Th18;
  A c= A\/A9 by XBOOLE_1:7;
  then [A\/A9, A] in F by A8;
  then [A\/A9, B] in F by A7,A10,Th18;
  hence thesis by A9,A12;
end;
