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 Th6:
  for E being non empty set, f being Function of E, E, c being
  Element of Class =_f, e being Element of c holds f.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;
  dom f = E by FUNCT_2:def 1;
  then
A1: f.e in dom f \/ rng f by XBOOLE_0:def 3;
  ex x9 being object st x9 in E & c = Class(=_f, x9) by EQREL_1:def 3;
  then
A2: c = Class(=_f, e) by EQREL_1:23;
  iter(f, 1).e = f.e by FUNCT_7:70
    .= id(field f).(f.e) by A1,FUNCT_1:17
    .= iter(f, 0).(f.e) by FUNCT_7:68;
  then [f.e,e] in =_f by Def7;
  hence thesis by A2,EQREL_1:19;
end;
