
theorem :: APP2:
  for X being set, F being Dependency-set of X holds F is (F1) (F2) (F3)
  (F4) iff F is (DC2) (DC5) (DC6)
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 (DC2) by A1;
    thus F is (DC5)
    proof
      let A, B, C, D be Subset of X such that
A5:   [A, B] in F and
A6:   [B\/C, D] in F;
      [C, C] in F by A1;
      then [A\/C, B\/C] in F by A4,A5;
      hence thesis by A2,A6,Th18;
    end;
    thus F is (DC6)
    proof
      let A, B, C be Subset of X such that
A7:   [A, B] in F;
      A c= A\/C by XBOOLE_1:7;
      then [A\/C, A] in F by A1,A3,Def15;
      hence thesis by A2,A7,Th18;
    end;
  end;
  assume that
A8: F is (DC2) and
A9: F is (DC5) and
A10: F is (DC6);
  thus F is (F1) by A8;
A11: now
    let A, B, C be Subset of X such that
A12: [A, B] in F and
A13: [B, C] in F;
    [B\/A, C] in F by A10,A13;
    then [A\/A, C] in F by A9,A12;
    hence [A, C] in F;
  end;
  hence F is (F2) by Th18;
  thus F is (F3)
  proof
    let A, B, A9, B9 be Subset of X such that
A14: [A, B] in F and
A15: [A, B] >= [A9, B9];
    A c= A9 by A15;
    then A9 = A\/(A9\A) by XBOOLE_1:45;
    then
A16: [A9, B] in F by A10,A14;
    B9 c= B by A15;
    then
A17: B = B9\/(B\B9) by XBOOLE_1:45;
    [B9, B9] in F by A8;
    then [ B, B9] in F by A10,A17;
    hence thesis by A11,A16;
  end;
  let A, B, A9, B9 be Subset of X such that
A18: [A, B] in F and
A19: [A9, B9] in F;
  [B\/B9, B\/B9] in F by A8;
  then [A\/B9, B\/B9] in F by A9,A18;
  hence thesis by A9,A19;
end;
