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 Th77:
  for A,B,C,D,E,F,J,M,N being set, h being Function, A9,B9,C9,D9,
E9,F9,J9,M9,N9 being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E
.--> E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .-->
  A9) holds dom h = {A,B,C,D,E,F,J,M,N}
proof
  let A,B,C,D,E,F,J,M,N be set;
  let h be Function;
  let A9,B9,C9,D9,E9,F9,J9,M9,N9 be set;
  assume
A1: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .--> A9);
A2: dom (A .--> A9) = {A};
  dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
  F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)) = {N,B,C,D,E,F,J,M} by Th63
    .= {N} \/ {B,C,D,E,F,J,M} by ENUMSET1:22
    .= {B,C,D,E,F,J,M,N} by ENUMSET1:28;
  then
  dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +*(A .--> A9)) = {B,C,D,E,F,J,
  M,N} \/ {A} by A2,FUNCT_4:def 1
    .= {A,B,C,D,E,F,J,M,N} by ENUMSET1:77;
  hence thesis by A1;
end;
