reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A,B,C,D,E,F for a_partition of Y;

theorem Th16:
  for A,B,C,D being object,h being Function,
   A9,B9,C9,D9 being object st
h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9) holds dom h = {A,B
  ,C,D}
proof
  let A,B,C,D be object;
  let h be Function;
  let A9,B9,C9,D9 be object;
  assume
A1: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9);
  dom ((B .--> B9) +* (C .--> C9)) = dom (B .--> B9) \/ dom (C .--> C9) by
FUNCT_4:def 1;
  then
A2: dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9)) = dom (B .--> B9) \/ dom
  (C .--> C9) \/ dom (D .--> D9) by FUNCT_4:def 1;
  dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9)) = {B} \/
  dom (C .--> C9) \/ dom (D .--> D9) \/ dom (A .--> A9) by A2,FUNCT_4:def 1
    .= {B} \/ {C} \/ dom (D .--> D9) \/ dom (A .--> A9)
    .= {B} \/ {C} \/ {D} \/ dom (A .--> A9)
    .= {A} \/ (({B} \/ {C}) \/ {D})
    .= {A} \/ ({B,C} \/ {D}) by ENUMSET1:1
    .= {A} \/ {B,C,D} by ENUMSET1:3
    .= {A,B,C,D} by ENUMSET1:4;
  hence thesis by A1;
end;
