reserve x, y, z, E, E1, E2, E3 for set,
  sE for Subset-Family of E,
  f for Function of E, E,
  k, l, m, n for Nat;

theorem
  for A being non empty set, f being Function of A,A, x being Element of
  A holds iter(f,n+1).x = f.(iter(f,n).x)
proof
  let A be non empty set, f be Function of A,A, x be Element of A;
  thus iter(f,n+1).x = (f*iter(f,n)).x by FUNCT_7:71
    .= f.(iter(f,n).x) by FUNCT_2:15;
end;
