let a be Real; for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = a ^2 ) & f2 = #Z 2 holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
let Z be open Subset of REAL; for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = a ^2 ) & f2 = #Z 2 holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
let f1, f2 be PartFunc of REAL,REAL; ( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = a ^2 ) & f2 = #Z 2 implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) ) )
assume that
A1:
( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = a ^2 ) )
and
A2:
f2 = #Z 2
; ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
A3:
( Z c= dom (f1 + (1 (#) f2)) & ( for x being Real st x in Z holds
f1 . x = (a ^2) + (0 * x) ) )
by A1, RFUNCT_1:21;
A4:
for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x
f1 + (1 (#) f2) is_differentiable_on Z
by A2, A3, Th12;
hence
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
by A4, RFUNCT_1:21; verum