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

theorem Th37:
  for A,B,C,D,E,F being set, h being Function, A9,B9,C9,D9,E9,F9
being set st A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B<>E & B<>F & C<>
  D & C<>E & C<>F & D<>E & D<>F & E<>F & h = (B .--> B9) +* (C .--> C9) +* (D
.--> D9) +* (E .--> E9) +* (F .--> F9) +* (A .--> A9) holds h.A = A9 & h.B = B9
  & h.C = C9 & h.D = D9 & h.E = E9 & h.F = F9
proof
  let A,B,C,D,E,F be set;
  let h be Function;
  let A9,B9,C9,D9,E9,F9 be set;
  assume that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: A<>E and
A5: A<>F and
A6: B<>C & B<>D & B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F and
A7: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (A .--> A9);
  A in dom (A .--> A9) by TARSKI:def 1;
  then
A9: h.A = (A .--> A9).A by A7,FUNCT_4:13;
  not C in dom (A .--> A9) by A2,TARSKI:def 1;
  then
A10: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9)).C by A7,FUNCT_4:11;
  not F in dom (A .--> A9) by A5,TARSKI:def 1;
  then
A11: h.F=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9)).F by A7,FUNCT_4:11
    .= F9 by A6,Th26;
  not E in dom (A .--> A9) by A4,TARSKI:def 1;
  then
A12: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9)).E by A7,FUNCT_4:11
    .= E9 by A6,Th26;
  not D in dom (A .--> A9) by A3,TARSKI:def 1;
  then
A13: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9)).D by A7,FUNCT_4:11
    .= D9 by A6,Th26;
  not B in dom (A .--> A9) by A1,TARSKI:def 1;
  then h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9)).B by A7,FUNCT_4:11
    .= B9 by A6,Th26;
  hence thesis by A6,A9,A10,A13,A12,A11,Th26,FUNCOP_1:72;
end;
