
theorem Th25:
  for I being non empty set
  for A, B being ManySortedSet of I
  for f being ManySortedSet of I
  holds f is ManySortedFunction of A,B
  iff (for i being Element of I holds f.i is Function of A.i, B.i)
proof
  let I be non empty set;
  let A, B be ManySortedSet of I;
  let f be ManySortedSet of I;
  thus f is ManySortedFunction of A,B
  implies (for i being Element of I holds f.i is Function of A.i, B.i)
    by PBOOLE:def 15;
  assume for i being Element of I holds f.i is Function of A.i, B.i;
  then for i being object st i in I holds
  f.i is Function of (A.i), (B.i);
  hence f is ManySortedFunction of A,B by PBOOLE:def 15;
end;
