
theorem MssRng:
  for I being non empty set
  for M being ManySortedSet of I
  for y being object
  holds y in rng M iff (ex i being Element of I st y = M.i)
proof
  let I be non empty set;
  let M be ManySortedSet of I;
  let y be object;
  hereby
    assume y in rng M;
    then consider i0 being object such that
    A1: i0 in dom M & y = M.i0 by FUNCT_1:def 3;
    reconsider i=i0 as Element of I by A1;
    take i;
    thus y = M.i by A1;
  end;
  given i being Element of I such that
  A2: y = M.i;
  dom M = I by PARTFUN1:def 2;
  hence y in rng M by A2, FUNCT_1:3;
end;
