let f1, f2 be PartFunc of COMPLEX,COMPLEX; :: thesis: for Z being open Subset of COMPLEX 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 Complex st x in Z holds
((f1 + f2) `| Z) /. x = (diff (f1,x)) + (diff (f2,x)) ) )
let Z be open Subset of COMPLEX; :: 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 Complex 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 Complex st x in Z holds
((f1 + f2) `| Z) /. x = (diff (f1,x)) + (diff (f2,x)) ) )

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

let x0 be Complex; :: 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, Th15;
hence f1 + f2 is_differentiable_in x0 by Th23; :: thesis: verum

end;

hence A3: f1 + f2 is_differentiable_on Z by A1, Th15; :: thesis: for x being Complex st x in Z holds
((f1 + f2) `| Z) /. x = (diff (f1,x)) + (diff (f2,x))

now :: thesis: for x being Complex st x in Z holds
((f1 + f2) `| Z) /. x = (diff (f1,x)) + (diff (f2,x))

let x be Complex; :: 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, Th15;
thus ((f1 + f2) `| Z) /. x = diff ((f1 + f2),x) by A3, A4, Def12
.= (diff (f1,x)) + (diff (f2,x)) by A5, Th23 ; :: thesis: verum

end;

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