let f1, f2 be PartFunc of REAL,REAL; :: thesis:
let Z be open Subset of REAL; :: thesis:
assume A1: ( Z c= dom ((f1 + f2) ^) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = #Z 2 ) ; :: thesis:
dom ((f1 + f2) ^) c= dom (f1 + f2) by RFUNCT_1:1;
then A2: Z c= dom (f1 + f2) by A1;
then A3: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) ) by A1, SIN_COS9:101;
A4: for x being Real st x in Z holds
(f1 + f2) . x <> 0 by A1, RFUNCT_1:3;
then A5: (f1 + f2) ^ is_differentiable_on Z by A3, FDIFF_2:22;
for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2))

let x be Real; :: thesis:
assume A6: x in Z ; :: thesis:
then A7: (f1 + f2) . x <> 0 by A1, RFUNCT_1:3;
A8: f1 + f2 is_differentiable_in x by A3, A6, FDIFF_1:9;
A9: f2 . x = x #Z 2 by A1, TAYLOR_1:def 1
.= x |^ 2 by PREPOWER:36 ;
A10: (f1 + f2) . x = (f1 . x) + (f2 . x) by A2, A6, VALUED_1:def 1
.= 1 + (x |^ 2) by A1, A6, A9 ;
(((f1 + f2) ^) `| Z) . x = diff (((f1 + f2) ^),x) by A5, A6, FDIFF_1:def 7
.= - ((diff ((f1 + f2),x)) / (((f1 + f2) . x) ^2)) by A7, A8, FDIFF_2:15
.= - ((((f1 + f2) `| Z) . x) / (((f1 + f2) . x) ^2)) by A3, A6, FDIFF_1:def 7
.= - ((2 * x) / ((1 + (x |^ 2)) ^2)) by A1, A2, A6, A10, SIN_COS9:101 ;
hence (((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ; :: thesis:

end;

hence ( (f1 + f2) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) ^) `| Z) . x = - ((2 * x) / ((1 + (x |^ 2)) ^2)) ) ) by A3, A4, FDIFF_2:22; :: thesis: