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 Th7:
  for E being non empty set, f being Function of E, E, c being
  Element of Class =_f, e being Element of c, n holds iter(f, n).e in c
proof
  let E be non empty set, f be Function of E, E;
  let c be Element of Class =_f, e be Element of c, n;
  dom f = E by FUNCT_2:def 1;
  then iter(f,n).e in dom f \/ rng f by XBOOLE_0:def 3;
  then iter(f, n).e = id(field f).(iter(f,n).e) by FUNCT_1:17
    .= iter(f, 0).(iter(f,n).e) by FUNCT_7:68;
  then
A1: [iter(f,n).e,e] in =_f by Def7;
  ex x9 being object st x9 in E & c = Class(=_f, x9) by EQREL_1:def 3;
  then c = Class(=_f, e) by EQREL_1:23;
  hence thesis by A1,EQREL_1:19;
end;
