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

theorem
for X be Subset of REAL m, f be PartFunc of REAL m,REAL, r be Real
  st X is open & f is_partial_differentiable_up_to_order i,X
 holds r(#)f is_partial_differentiable_up_to_order i,X
proof
  let Z be Subset of REAL m, f be PartFunc of REAL m,REAL, r be Real;
  assume A1: Z is open & f is_partial_differentiable_up_to_order i,Z;
  let I be non empty FinSequence of NAT;
  assume A2:len I <= i & rng I c= Seg m; then
  f is_partial_differentiable_on Z,I by A1;
  hence thesis by A1,A2,Th79;
end;
