reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A,B,C,D,E,F for a_partition of Y;
reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A, B, C, D, E, F, J, M for a_partition of Y,
  x,x1,x2,x3,x4,x5,x6,x7,x8,x9 for set;

theorem Th50:
  for A,B,C,D,E,F,J being set, h being Function, A9,B9,C9,D9,E9,F9
  ,J9 being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)
  +* (F .--> F9) +* (J .--> J9) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J}
proof
  let A,B,C,D,E,F,J be set;
  let h be Function;
  let A9,B9,C9,D9,E9,F9,J9 be set;
  assume h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (A .--> A9);
  then
  dom h = dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +*
  (F .--> F9) +* (J .--> J9)) \/ dom (A .--> A9) by FUNCT_4:def 1
    .= {J,B,C,D,E,F} \/ dom (A .--> A9) by Th38
    .= ({B,C,D,E,F} \/ {J}) \/ {A} by ENUMSET1:11
    .= {B,C,D,E,F,J} \/ {A} by ENUMSET1:15
    .= {A,B,C,D,E,F,J} by ENUMSET1:16;
  hence thesis;
end;
