
theorem Th19:
  for X being set holds Dependencies X is (F1) (F2) (F3) (F4)
proof
  let X be set;
  set D = Dependencies X;
  thus D is (F1);
  D = nabla bool X by EQREL_1:def 1;
  then
A1: field D = bool X by ORDERS_1:12;
  thus D is (F2)
  proof
    let x, y, z be object;
    assume that
A2: x in field D and
    y in field D and
A3: z in field D and
    [x,y] in D and
    [y,z] in D;
    thus thesis by A1,A2,A3,ZFMISC_1:def 2;
  end;
  thus D is (F3);
  thus D is (F4);
end;
