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 Th64:
  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 h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .-->
E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (A .--> A9) holds rng h = {
  h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M}
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 h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (A .--> A9);
  then
A1: dom h = {A,B,C,D,E,F,J,M} by Th63;
  then B in dom h by ENUMSET1:def 6;
  then
A2: h.B in rng h by FUNCT_1:def 3;
  M in dom h by A1,ENUMSET1:def 6;
  then
A3: h.M in rng h by FUNCT_1:def 3;
  J in dom h by A1,ENUMSET1:def 6;
  then
A4: h.J in rng h by FUNCT_1:def 3;
  F in dom h by A1,ENUMSET1:def 6;
  then
A5: h.F in rng h by FUNCT_1:def 3;
  E in dom h by A1,ENUMSET1:def 6;
  then
A6: h.E in rng h by FUNCT_1:def 3;
A7: rng h c= {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M}
  proof
    let t be object;
    assume t in rng h;
    then consider x1 being object such that
A8: x1 in dom h and
A9: t = h.x1 by FUNCT_1:def 3;
    x1=A or x1=B or x1=C or x1=D or x1=E or x1=F or x1=J or x1=M by A1,A8,
ENUMSET1:def 6;
    hence thesis by A9,ENUMSET1:def 6;
  end;
  D in dom h by A1,ENUMSET1:def 6;
  then
A10: h.D in rng h by FUNCT_1:def 3;
  C in dom h by A1,ENUMSET1:def 6;
  then
A11: h.C in rng h by FUNCT_1:def 3;
  A in dom h by A1,ENUMSET1:def 6;
  then
A12: h.A in rng h by FUNCT_1:def 3;
  {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M} c= rng h
  by A12,A2,A11,A10,A6,A5,A4,A3,ENUMSET1:def 6;
  hence thesis by A7,XBOOLE_0:def 10;
end;
