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) and
A2: for x being Real st x in Z holds
f1 . x = a ^2 and
A3: 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 ) )
A4: Z c= dom (f1 + (1 (#) f2)) by A1, RFUNCT_1:33;
A5: for x being Real st x in Z holds
f1 . x = (a ^2 ) + (0 * x) by A2;
then A6: ( f1 + (1 (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (1 (#) f2)) `| Z) . x = 0 + ((2 * 1) * x) ) ) by A3, A4, Th12;
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 A7: x in Z ; :: thesis: ((f1 + f2) `| Z) . x = 2 * x
((f1 + f2) `| Z) . x = ((f1 + (1 (#) f2)) `| Z) . x by RFUNCT_1:33
.= 0 + ((2 * 1) * x) by A3, A4, A5, A7, Th12
.= 2 * x ;
hence ((f1 + f2) `| Z) . x = 2 * x ; :: thesis: verum

end;

hence ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) ) by A6, RFUNCT_1:33; :: thesis: verum