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 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 ) & 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 ) & 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 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: for x being Real st x in Z holds
f2 is_differentiable_in x by A3, TAYLOR_1:2;
A5: Z c= (dom f1) /\ (dom f2) by A1, VALUED_1:def 1;
then A6: Z c= dom f1 by XBOOLE_1:18;
A7: for x being Real st x in Z holds
f1 . x = (0 * x) + a by A2;
then A8: f1 is_differentiable_on Z by A6, FDIFF_1:23;
Z c= dom f2 by A5, XBOOLE_1:18;
then A9: f2 is_differentiable_on Z by A4, FDIFF_1:9;
A10: for x being Real st x in Z holds
(f2 `| Z) . x = 2 * (x #Z (2 - 1))

proof

let x be Real; :: thesis: ( x in Z implies (f2 `| Z) . x = 2 * (x #Z (2 - 1)) )
assume A11: x in Z ; :: thesis: (f2 `| Z) . x = 2 * (x #Z (2 - 1))
diff (f2,x) = 2 * (x #Z (2 - 1)) by A3, TAYLOR_1:2;
hence (f2 `| Z) . x = 2 * (x #Z (2 - 1)) by A9, A11, FDIFF_1:def 7; :: thesis: verum

end;

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 A12: x in Z ; :: thesis: ((f1 + f2) `| Z) . x = 2 * x
then ((f1 + f2) `| Z) . x = (diff (f1,x)) + (diff (f2,x)) by A1, A8, A9, FDIFF_1:18;
hence ((f1 + f2) `| Z) . x = ((f1 `| Z) . x) + (diff (f2,x)) by A8, A12, FDIFF_1:def 7
.= ((f1 `| Z) . x) + ((f2 `| Z) . x) by A9, A12, FDIFF_1:def 7
.= 0 + ((f2 `| Z) . x) by A6, A7, A12, FDIFF_1:23
.= 2 * (x #Z (2 - 1)) by A10, A12
.= 2 * (x |^ 1) by PREPOWER:36
.= 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 A1, A8, A9, FDIFF_1:18; :: thesis: verum