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 Th62:
  for A,B,C,D,E,F,J,M being set, h being Function, A9,B9,C9,D9,E9,
F9,J9,M9 being set st A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B<>C & B<>D & B
<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C<>M & D<>E & D<>F & D<>
J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M & h = (B .--> B9) +* (C .-->
C9) +* (D .--> D9) +* (E .--> E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9)
+* (A .--> A9) holds h.B = B9 & h.C = C9 & h.D = D9 & h.E = E9 & h.F = F9 & h.J
  = J9
proof
  let A,B,C,D,E,F,J,M be set;
  let h be Function;
  let A9,B9,C9,D9,E9,F9,J9,M9 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: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J
  & C<> M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
  and
A8: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (A .--> A9);
  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) +* (J .--> J9) +* (M .--> M9)).C by A8,FUNCT_4:11;
  not J in dom (A .--> A9) by A6,TARSKI:def 1;
  then
A11: h.J=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9)).J by A8,FUNCT_4:11
    .= J9 by A7,Th49;
  not F in dom (A .--> A9) by A5,TARSKI:def 1;
  then
A12: h.F=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9)).F by A8,FUNCT_4:11
    .= F9 by A7,Th49;
  not E in dom (A .--> A9) by A4,TARSKI:def 1;
  then
A13: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9)).E by A8,FUNCT_4:11
    .= E9 by A7,Th49;
  not D in dom (A .--> A9) by A3,TARSKI:def 1;
  then
A14: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9)).D by A8,FUNCT_4:11
    .= D9 by A7,Th49;
  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)).B by A8,FUNCT_4:11
    .= B9 by A7,Th49;
  hence thesis by A7,A10,A14,A13,A12,A11,Th49;
end;
