
theorem
  for I being set
  for F being ManySortedSet of I
  holds F is Group-yielding
  iff (for i being object st i in I holds F.i is Group)
proof
  let I be set;
  let F be ManySortedSet of I;
  hereby
    assume A1: F is Group-yielding;
    let i be object;
    assume i in I;
    then i in dom F by PARTFUN1:def 2;
    hence F.i is Group by A1, FUNCT_1:3;
  end;
  assume A2: for i being object st i in I holds F.i is Group;
  for y being object st y in rng F holds y is Group
  proof
    let y be object;
    assume y in rng F;
    then consider i being object such that
    A3: i in dom F & y = F.i by FUNCT_1:def 3;
    thus y is Group by A2, A3;
  end;
  hence F is Group-yielding;
end;
