
theorem Th9:
  for m be non zero Element of NAT, Z be non empty Subset of REAL m,
      f be PartFunc of REAL m,REAL, i,j be Nat,
      I be non empty FinSequence of NAT
        st f is_continuously_differentiable_up_to_order (i+j),Z
          & rng I c= Seg m & len I = j
      holds
        f`partial|(Z,I) is_continuously_differentiable_up_to_order i,Z
proof
  let m be non zero Element of NAT, Z be non empty Subset of REAL m,
      f be PartFunc of REAL m,REAL, i,j be Nat,
      I be non empty FinSequence of NAT;
  assume
A1: f is_continuously_differentiable_up_to_order (i+j),Z
      & rng I c= Seg m & len I = j;
  then
A2: Z c= dom f & f is_partial_differentiable_up_to_order (i+j),Z
     & for I be non empty FinSequence of NAT
       st len I <= (i+j) & rng I c= Seg m
        holds f`partial|(Z,I) is_continuous_on Z;
  A3:  f is_partial_differentiable_on Z,I by A2,A1,NAT_1:11;
  f is_partial_differentiable_on Z,I by A2,A1,NAT_1:11;
  hence Z c= dom (f`partial|(Z,I)) by Th1;
  thus f`partial|(Z,I) is_partial_differentiable_up_to_order i,Z
                                              by PDIFF_9:83,A1;
  let J be non empty FinSequence of NAT;
  assume
A4:len J <= i & rng J c= Seg m;
  reconsider G = I^J as non empty FinSequence of NAT;
A5:rng G = rng I \/ rng J by FINSEQ_1:31;
  len G = len I + len J by FINSEQ_1:22;
  then
    len G <= i+j & rng G c= Seg m by A4,A5,A1,XBOOLE_1:8,XREAL_1:6;
  then f`partial|(Z,G) is_continuous_on Z by A1;
  hence thesis by A3,Th7;
end;
