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

theorem Th15:
  for h being Function, A9,B9,C9,D9 being object st A<>B & A<>C & A<>
  D & B<>C & B<>D & C<>D & h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A
  .--> A9) holds h.B = B9 & h.C = C9 & h.D = D9
proof
  let h be Function;
  let A9,B9,C9,D9 be object;
  assume that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: B<>C and
A5: B<>D and
A6: C<>D and
A7: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9);
  not D in dom (A .--> A9) by A3,TARSKI:def 1;
  then
A9: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).D by A7,FUNCT_4:11;
  not C in dom (A .--> A9) by A2,TARSKI:def 1;
  then
A10: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).C by A7,FUNCT_4:11;
  not C in dom (D .--> D9) by A6,TARSKI:def 1;
  then
A12: h.C=((B .--> B9) +* (C .--> C9)).C by A10,FUNCT_4:11;
  not B in dom (A .--> A9) by A1,TARSKI:def 1;
  then
A13: h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).B by A7,FUNCT_4:11;
  not B in dom (D .--> D9) by A5,TARSKI:def 1;
  then
A14: h.B=((B .--> B9) +* (C .--> C9)).B by A13,FUNCT_4:11;
  not B in dom (C .--> C9) by A4,TARSKI:def 1;
  then h.B=(B .--> B9).B by A14,FUNCT_4:11;
  hence h.B = B9 by FUNCOP_1:72;
  C in dom (C .--> C9) by TARSKI:def 1;
  then h.C=(C .--> C9).C by A12,FUNCT_4:13;
  hence h.C = C9 by FUNCOP_1:72;
  D in dom (D .--> D9) by TARSKI:def 1;
  then h.D=(D .--> D9).D by A9,FUNCT_4:13;
  hence thesis by FUNCOP_1:72;
end;
