reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th12:
  for h being FinSequence of REAL holds
    h is one-to-one iff sort_a h is one-to-one
proof
  let h be FinSequence of REAL;
A1: h,(sort_a h) are_fiberwise_equipotent by RFINSEQ2:def 6;
  then ex H be Function st dom H = dom (sort_a h) & rng H = dom h & H is
  one-to-one & (sort_a h) = h*H by CLASSES1:77;
  hence h is one-to-one implies sort_a h is one-to-one;
  ex G be Function st dom G = dom (h) & rng G = dom (sort_a h) & G is
  one-to-one & (h) = (sort_a h) *G by A1,CLASSES1:77;
  hence sort_a h is one-to-one implies h is one-to-one;
end;
