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

theorem Th38:
  for A,B,C,D,E,F being set, h being Function, A9,B9,C9,D9,E9,F9
being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (A .--> A9) holds dom h = {A,B,C,D,E,F}
proof
  let A,B,C,D,E,F be set;
  let h be Function;
  let A9,B9,C9,D9,E9,F9 be set;
  assume
A1: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (A .--> A9);
A2: dom (A .--> A9) = {A};
  dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
  F9)) = {F,B,C,D,E} by Th27
    .= {F} \/ {B,C,D,E} by ENUMSET1:7
    .= {B,C,D,E,F} by ENUMSET1:10;
  then
  dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
  F9) +*(A .--> A9)) = {B,C,D,E,F} \/ {A} by A2,FUNCT_4:def 1
    .= {A,B,C,D,E,F} by ENUMSET1:11;
  hence thesis by A1;
end;
