let F be RealNormSpace; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL, the carrier of F st Z c= dom (f1 + f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) ) )
let Z be open Subset of REAL; :: thesis: for f1, f2 being PartFunc of REAL, the carrier of F st Z c= dom (f1 + f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) ) )
let f1, f2 be PartFunc of REAL, the carrier of F; :: thesis: ( Z c= dom (f1 + f2) & f1 is_differentiable_on Z & f2 is_differentiable_on Z implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) ) ) )
assume that
A1: Z c= dom (f1 + f2) and
A2: ( f1 is_differentiable_on Z & f2 is_differentiable_on Z ) ; :: thesis: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) ) )

now :: thesis: for x0 being Real st x0 in Z holds
f1 + f2 is_differentiable_in x0

let x0 be Real; :: thesis: ( x0 in Z implies f1 + f2 is_differentiable_in x0 )
assume x0 in Z ; :: thesis: f1 + f2 is_differentiable_in x0
then ( f1 is_differentiable_in x0 & f2 is_differentiable_in x0 ) by A2, Th10;
hence f1 + f2 is_differentiable_in x0 by Th14; :: thesis: verum

end;

hence A3: f1 + f2 is_differentiable_on Z by A1, Th10; :: thesis: for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x))
let x be Real; :: thesis: ( x in Z implies ((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) )
assume A4: x in Z ; :: thesis: ((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x))
then A5: ( f1 is_differentiable_in x & f2 is_differentiable_in x ) by A2, Th10;
thus ((f1 + f2) `| Z) . x = diff ((f1 + f2),x) by A3, A4, Def6
.= (diff (f1,x)) + (diff (f2,x)) by A5, Th14 ; :: thesis: verum