
theorem Th18:
  for X being set, F being Dependency-set of X holds F is (F2) iff
  for A, B, C being Subset of X st [A, B] in F & [B, C] in F holds [A, C] in F
proof
  let X be set, F be Dependency-set of X;
  hereby
    assume F is (F2);
    then
A1: F is_transitive_in field F;
    let A, B, C be Subset of X;
    assume that
A2: [A, B] in F and
A3: [B, C] in F;
A4: B in field F by A2,RELAT_1:15;
A5: C in field F by A3,RELAT_1:15;
    A in field F by A2,RELAT_1:15;
    hence [A, C] in F by A1,A2,A3,A4,A5;
  end;
  assume
A6: for A,B,C being Subset of X st [A, B] in F & [B, C] in F holds [A, C
  ] in F;
  let x, y, z be object such that
A7: x in field F and
A8: y in field F and
A9: z in field F and
A10: [x,y] in F and
A11: [y,z] in F;
  field F c= bool X\/bool X by RELSET_1:8;
  then reconsider A = x, B = y, C = z as Subset of X by A7,A8,A9;
A12: [B, C] in F by A11;
  [A, B] in F by A10;
  hence thesis by A6,A12;
end;
