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 ) & f2 = #Z 3 holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 3 * (x |^ 2) ) )
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 ) & f2 = #Z 3 holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 3 * (x |^ 2) ) )
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 ) & f2 = #Z 3 implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 3 * (x |^ 2) ) ) )
assume A1: ( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = a ) & f2 = #Z 3 ) ; :: thesis: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 3 * (x |^ 2) ) )
then Z c= (dom f1) /\ (dom f2) by VALUED_1:def 1;
then A2: ( Z c= dom f1 & Z c= dom f2 ) by XBOOLE_1:18;
A3: for x being Real st x in Z holds
f1 . x = (0 * x) + a by A1;
then A4: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 0 ) ) by A2, FDIFF_1:31;
for x being Real st x in Z holds
f2 is_differentiable_in x by A1, TAYLOR_1:2;
then A5: f2 is_differentiable_on Z by A2, FDIFF_1:16;
A6: for x being Real st x in Z holds
(f2 `| Z) . x = 3 * (x #Z (3 - 1))

proof

let x be Real; :: thesis: ( x in Z implies (f2 `| Z) . x = 3 * (x #Z (3 - 1)) )
assume A7: x in Z ; :: thesis: (f2 `| Z) . x = 3 * (x #Z (3 - 1))
diff f2,x = 3 * (x #Z (3 - 1)) by A1, TAYLOR_1:2;
hence (f2 `| Z) . x = 3 * (x #Z (3 - 1)) by A5, A7, FDIFF_1:def 8; :: thesis: verum

end;

for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 3 * (x |^ 2)

proof

let x be Real; :: thesis: ( x in Z implies ((f1 + f2) `| Z) . x = 3 * (x |^ 2) )
assume A8: x in Z ; :: thesis: ((f1 + f2) `| Z) . x = 3 * (x |^ 2)
then ((f1 + f2) `| Z) . x = (diff f1,x) + (diff f2,x) by A1, A4, A5, FDIFF_1:26;
hence ((f1 + f2) `| Z) . x = ((f1 `| Z) . x) + (diff f2,x) by A4, A8, FDIFF_1:def 8
.= ((f1 `| Z) . x) + ((f2 `| Z) . x) by A5, A8, FDIFF_1:def 8
.= 0 + ((f2 `| Z) . x) by A2, A3, A8, FDIFF_1:31
.= 3 * (x #Z (3 - 1)) by A6, A8
.= 3 * (x |^ 2) by PREPOWER:46 ;
:: thesis: verum

end;

hence ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 3 * (x |^ 2) ) ) by A1, A4, A5, FDIFF_1:26; :: thesis: verum