reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LM215:
  for X, Y, W be RealNormSpace,
  z be Point of [:X,Y:],
  f1, f2 be PartFunc of [:X,Y:], W
  st f1 is_partial_differentiable_in`1 z &
  f2 is_partial_differentiable_in`1 z holds
  (f1+f2) is_partial_differentiable_in`1 z &
  partdiff`1(f1+f2,z) = partdiff`1(f1,z)+partdiff`1(f2,z) &
  (f1-f2) is_partial_differentiable_in`1 z &
  partdiff`1(f1-f2,z) = partdiff`1(f1,z)-partdiff`1(f2,z)
  proof
    let X, Y, W be RealNormSpace,
    z be Point of [:X,Y:],
    f1,f2 be PartFunc of [:X,Y:],W;
    assume
A1: f1 is_partial_differentiable_in`1 z &
    f2 is_partial_differentiable_in`1 z;
    f1*reproj1(z) + f2*reproj1(z) is_differentiable_in z`1 by NDIFF_1:35,A1;
    hence (f1+f2) is_partial_differentiable_in`1 z by Th26;
    thus partdiff`1(f1+f2,z) =
         diff(f1*reproj1(z)+f2*reproj1(z),z`1) by Th26
       .= partdiff`1(f1,z)+partdiff`1(f2,z) by NDIFF_1:35,A1;
    f1*reproj1(z) - f2*reproj1(z) is_differentiable_in z`1 by NDIFF_1:36,A1;
    hence (f1-f2) is_partial_differentiable_in`1 z by Th26;
    thus partdiff`1(f1-f2,z) =
         diff(f1*reproj1(z)-f2*reproj1(z),z`1) by Th26
      .= partdiff`1(f1,z)-partdiff`1(f2,z) by NDIFF_1:36,A1;
   end;
