reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;

theorem
  for A be non-empty ManySortedSet of I for F be ManySortedFunction of A
  , EmptyMS I holds F = EmptyMS I
proof
  let A be non-empty ManySortedSet of I;
  let F be ManySortedFunction of A, EmptyMS I;
  now
    let i be object;
    assume i in I;
    then reconsider f = F.i as Function of A.i, EmptyMS I.i by PBOOLE:def 15;
    f = {};
    hence F.i = EmptyMS I.i;
  end;
  hence thesis;
end;
