let a be Real; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ; :: thesis: ( 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

proof

let x be Real; :: thesis: ( x in Z implies ((f1 + f2) `| Z) . x = 2 * x )
assume A5: x in Z ; :: thesis: ((f1 + f2) `| Z) . x = 2 * x
((f1 + f2) `| Z) . x = ((f1 + (1 (#) f2)) `| Z) . x by RFUNCT_1:21
.= 0 + ((2 * 1) * x) by A2, A3, A5, Th12
.= 2 * x ;
hence ((f1 + f2) `| Z) . x = 2 * x ; :: thesis: verum

end;

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; :: thesis: verum