reserve x for object, X, F for set;
reserve c, c1, c2, c3 for Complex,
  e, e1, e2, e3 for ExtReal,
  r , r1, r2, r3 for Real,
  w, w1, w2, w3 for Rational,
  i, i1, i2, i3 for Integer,
  n, n1, n2, n3 for Nat;

theorem
  for X being rational-membered set st for w holds w in X holds X = RAT
proof
  let X be rational-membered set such that
A1: for w holds w in X;
  thus X c= RAT by Th4;
  let e be object;
  assume e in RAT;
  hence thesis by A1;
end;
