reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th83:
for i,j being Nat
for f be PartFunc of REAL m,REAL,
    I be non empty FinSequence of NAT
 st f is_partial_differentiable_up_to_order (i+j),Z
  & rng I c= Seg m & len I = j
holds
  f`partial|(Z,I) is_partial_differentiable_up_to_order i,Z
proof let i,j be Nat;
   let f be PartFunc of REAL m,REAL,
       I be non empty FinSequence of NAT;
   assume A1: f is_partial_differentiable_up_to_order (i+j),Z
            & rng I c= Seg m & len I = j;
   let J be non empty FinSequence of NAT;
   assume A2: len J <= i & rng J c= Seg m;
   reconsider G = I^J as non empty FinSequence of NAT;
A3: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 A2,A3,A1,XBOOLE_1:8,XREAL_1:6; then
   f is_partial_differentiable_on Z,G by A1;
   hence thesis by Th80;
end;
