let S, T be non trivial RealNormSpace; :: thesis:
let Z be Subset of S; :: thesis:
assume A1: Z is open ; :: thesis:
let f1, f2 be PartFunc of S,T; :: thesis:
assume that
A2: Z c= dom (f1 + f2) and
A3: ( f1 is_differentiable_on Z & f2 is_differentiable_on Z ) ; :: thesis:

let x0 be Point of S; :: thesis:
assume x0 in Z ; :: thesis:
then ( f1 is_differentiable_in x0 & f2 is_differentiable_in x0 ) by A1, A3, Th36;
hence f1 + f2 is_differentiable_in x0 by Th40; :: thesis:

end;

hence A4: f1 + f2 is_differentiable_on Z by A1, A2, Th36; :: thesis:

let x be Point of S; :: thesis:
assume A5: x in Z ; :: thesis:
then A6: ( f1 is_differentiable_in x & f2 is_differentiable_in x ) by A1, A3, Th36;
thus ((f1 + f2) `| Z) /. x = diff ((f1 + f2),x) by A4, A5, Def9
.= (diff (f1,x)) + (diff (f2,x)) by A6, Th40 ; :: thesis:

end;

hence for x being Point of S st x in Z holds
((f1 + f2) `| Z) /. x = (diff (f1,x)) + (diff (f2,x)) ; :: thesis: