
theorem Th27:
  for X being set, F being Dependency-set of X, A, B being Subset
of X holds A ^|^ B, F iff [A, B] in F & not ex A9, B9 being Subset of X st [A9,
  B9] in F & (A <> A9 or B <> B9) & [A, B] <= [A9, B9]
proof
  let X be set, F be Dependency-set of X, x, y be Subset of X;
  set DOX = Dependencies-Order X;
  hereby
    assume x ^|^ y, F;
    then
A1: [x, y] in Maximal_wrt F;
    hence [x, y] in F;
A2: [x, y] is_maximal_wrt F, DOX by A1,Def1;
    given x9, y9 being Subset of X such that
A3: [x9, y9] in F and
A4: x <> x9 or y <> y9 and
A5: [x, y] <= [x9,y9];
A6: [[x,y],[x9,y9]] in DOX by A5;
    [x,y] <> [x9,y9] by A4,XTUPLE_0:1;
    hence contradiction by A2,A3,A6;
  end;
  assume that
A7: [x, y] in F and
A8: not ex x9, y9 being Subset of X st [x9, y9] in F & (x <> x9 or y <>
  y9) & [x, y] <= [x9,y9];
  [x,y] is_maximal_wrt F, DOX
  proof
    thus [x,y] in F by A7;
    given z being set such that
A9: z in F and
A10: z <> [x,y] and
A11: [[x, y],z] in DOX;
    consider x9,y9 being object such that
A12: z = [x9,y9] and
A13: x9 in bool X and
A14: y9 in bool X by A9,RELSET_1:2;
    consider e, f being Dependency of X such that
A15: [[x, y],z] = [e, f] and
A16: e <= f by A11;
A17: e = [x,y] by A15,XTUPLE_0:1;
A18: f = z by A15,XTUPLE_0:1;
    reconsider x9, y9 as Subset of X by A13,A14;
    x <> x9 or y <> y9 by A10,A12;
    hence contradiction by A8,A9,A12,A13,A14,A16,A17,A18;
  end;
  then [x,y] in Maximal_wrt F by Def1;
  hence thesis;
end;
