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,g be PartFunc of REAL m,REAL
 st X is open & f is_partial_differentiable_up_to_order i,X
  & g is_partial_differentiable_up_to_order i,X
holds f+g is_partial_differentiable_up_to_order i,X
    & f-g is_partial_differentiable_up_to_order i,X
proof
  let Z be Subset of REAL m, f,g be PartFunc of REAL m,REAL;
  assume A1: Z is open & f is_partial_differentiable_up_to_order i,Z
  & g is_partial_differentiable_up_to_order i,Z;
  hereby 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
    & g is_partial_differentiable_on Z,I by A1;
    hence f+g is_partial_differentiable_on Z,I by A1,A2,Th75;
  end;
  let I be non empty FinSequence of NAT;
  assume A3:len I <= i & rng I c= Seg m; then
  f is_partial_differentiable_on Z,I
  & g is_partial_differentiable_on Z,I by A1;
  hence thesis by A1,A3,Th77;
end;
