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 Th76:
  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 A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & A
<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F & C<>
J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M & E<>N
& F<>J & F<>M & F<>N & J<>M & J<>N & M<>N & h = (B .--> B9) +* (C .--> C9) +* (
  D .--> D9) +* (E .--> E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N
.--> N9) +* (A .--> A9) holds h.A = A9 & h.B = B9 & h.C = C9 & h.D = D9 & h.E =
  E9 & h.F = F9 & h.J = J9 & h.M = M9 & h.N = N9
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 that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: A<>E and
A5: A<>F and
A6: A<>J and
A7: A<>M and
A8: A<>N and
A9: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F
& C<> J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M &
  E<>N & F<>J & F<>M & F<>N & J<>M & J<>N and
A10: M<>N and
A11: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .--> A9);
  A in dom (A .--> A9) by TARSKI:def 1;
  then
A13: h.A = (A .--> A9).A by A11,FUNCT_4:13;
  not E in dom (A .--> A9) by A4,TARSKI:def 1;
  then
A14: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).E by A11,FUNCT_4:11
    .= E9 by A9,Th62;
  not N in dom (A .--> A9) by A8,TARSKI:def 1;
  then
A15: h.N=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).N by A11,FUNCT_4:11
    .= N9 by FUNCT_7:94;
  not D in dom (A .--> A9) by A3,TARSKI:def 1;
  then
A16: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).D by A11,FUNCT_4:11
    .= D9 by A9,Th62;
  not C in dom (A .--> A9) by A2,TARSKI:def 1;
  then
A17: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).C by A11,FUNCT_4:11;
  not J in dom (A .--> A9) by A6,TARSKI:def 1;
  then
A18: h.J=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).J by A11,FUNCT_4:11
    .= J9 by A9,Th62;
  not F in dom (A .--> A9) by A5,TARSKI:def 1;
  then
A19: h.F=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).F by A11,FUNCT_4:11
    .= F9 by A9,Th62;
  not M in dom (A .--> A9) by A7,TARSKI:def 1;
  then
A20: h.M=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).M by A11,FUNCT_4:11
    .= M9 by A10,Lm1;
  not B in dom (A .--> A9) by A1,TARSKI:def 1;
  then h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).B by A11,FUNCT_4:11
    .= B9 by A9,Th62;
  hence thesis by A9,A13,A17,A16,A14,A19,A18,A20,A15,Th62,FUNCOP_1:72;
end;
