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

theorem Th26:
  for A,B,C,D,E being set, h being Function, A9,B9,C9,D9,E9 being
set st A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E & h
= (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A .--> A9) holds
  h.A = A9 & h.B = B9 & h.C = C9 & h.D = D9 & h.E = E9
proof
  let A,B,C,D,E be set;
  let h be Function;
  let A9,B9,C9,D9,E9 be set;
  assume that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: A<>E and
A5: B<>C and
A6: B<>D and
A7: B<>E and
A8: C<>D and
A9: C<>E and
A10: D<>E and
A11: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A
  .--> A9);
  A in dom (A .--> A9) by TARSKI:def 1;
  then
A13: h.A = (A .--> A9).A by A11,FUNCT_4:13;
  not C in dom (A .--> A9) by A2,TARSKI:def 1;
  then
A14: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)).C by A11,
FUNCT_4:11;
  not B in dom (D .--> D9) by A6,TARSKI:def 1;
  then
A16: ((B .--> B9) +* (C .--> C9) +* (D .--> D9)).B= ((B .--> B9) +* (C .-->
  C9)).B by FUNCT_4:11;
  not E in dom (A .--> A9) by A4,TARSKI:def 1;
  then
A17: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)).E by A11,
FUNCT_4:11;
  E in dom (E .--> E9) by TARSKI:def 1;
  then
A19: h.E=(E .--> E9).E by A17,FUNCT_4:13;
  not C in dom (D .--> D9) by A8,TARSKI:def 1;
  then
A20: ((B .--> B9) +* (C .--> C9) +* (D .--> D9)).C= ((B .--> B9) +* (C .-->
  C9)).C by FUNCT_4:11;
  not C in dom (E .--> E9) by A9,TARSKI:def 1;
  then
A21: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).C by A14,FUNCT_4:11;
  C in dom (C .--> C9) by TARSKI:def 1;
  then
A23: h.C=(C .--> C9).C by A21,A20,FUNCT_4:13;
  not D in dom (A .--> A9) by A3,TARSKI:def 1;
  then
A24: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)).D by A11,
FUNCT_4:11;
  not D in dom (E .--> E9) by A10,TARSKI:def 1;
  then
A25: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).D by A24,FUNCT_4:11;
  D in dom (D .--> D9) by TARSKI:def 1;
  then
A26: h.D=(D .--> D9).D by A25,FUNCT_4:13;
  not B in dom (A .--> A9) by A1,TARSKI:def 1;
  then
A27: h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)).B by A11,
FUNCT_4:11;
  not B in dom (E .--> E9) by A7,TARSKI:def 1;
  then
A28: h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).B by A27,FUNCT_4:11;
  not B in dom (C .--> C9) by A5,TARSKI:def 1;
  then h.B=(B .--> B9).B by A28,A16,FUNCT_4:11;
  hence thesis by A13,A23,A26,A19,FUNCOP_1:72;
end;
