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 Th78:
  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 h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E
.--> E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .-->
  A9) holds rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N}
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 h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
  .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .--> A9);
  then
A1: dom h = {A,B,C,D,E,F,J,M,N} by Th77;
  then
A2: B in dom h by ENUMSET1:def 7;
A3: M in dom h by A1,ENUMSET1:def 7;
A4: J in dom h by A1,ENUMSET1:def 7;
A5: N in dom h by A1,ENUMSET1:def 7;
A6: D in dom h by A1,ENUMSET1:def 7;
A7: C in dom h by A1,ENUMSET1:def 7;
A8: rng h c= {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N}
  proof
    let t be object;
    assume t in rng h;
    then consider x1 being object such that
A9: x1 in dom h and
A10: t = h.x1 by FUNCT_1:def 3;
    now
      per cases by A1,A9,ENUMSET1:def 7;
      case
        x1=A;
        hence thesis by A10,ENUMSET1:def 7;
      end;
      case
        x1=B;
        hence thesis by A10,ENUMSET1:def 7;
      end;
      case
        x1=C;
        hence thesis by A10,ENUMSET1:def 7;
      end;
      case
        x1=D;
        hence thesis by A10,ENUMSET1:def 7;
      end;
      case
        x1=E;
        hence thesis by A10,ENUMSET1:def 7;
      end;
      case
        x1=F;
        hence thesis by A10,ENUMSET1:def 7;
      end;
      case
        x1=J;
        hence thesis by A10,ENUMSET1:def 7;
      end;
      case
        x1=M;
        hence thesis by A10,ENUMSET1:def 7;
      end;
      case
        x1=N;
        hence thesis by A10,ENUMSET1:def 7;
      end;
    end;
    hence thesis;
  end;
A11: F in dom h by A1,ENUMSET1:def 7;
A12: E in dom h by A1,ENUMSET1:def 7;
A13: A in dom h by A1,ENUMSET1:def 7;
  {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} c= rng h
  proof
    let t be object;
    assume
A14: t in {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N};
    now
      per cases by A14,ENUMSET1:def 7;
      case
        t=h.A;
        hence thesis by A13,FUNCT_1:def 3;
      end;
      case
        t=h.B;
        hence thesis by A2,FUNCT_1:def 3;
      end;
      case
        t=h.C;
        hence thesis by A7,FUNCT_1:def 3;
      end;
      case
        t=h.D;
        hence thesis by A6,FUNCT_1:def 3;
      end;
      case
        t=h.E;
        hence thesis by A12,FUNCT_1:def 3;
      end;
      case
        t=h.F;
        hence thesis by A11,FUNCT_1:def 3;
      end;
      case
        t=h.J;
        hence thesis by A4,FUNCT_1:def 3;
      end;
      case
        t=h.M;
        hence thesis by A3,FUNCT_1:def 3;
      end;
      case
        t=h.N;
        hence thesis by A5,FUNCT_1:def 3;
      end;
    end;
    hence thesis;
  end;
  hence thesis by A8,XBOOLE_0:def 10;
end;
