let r be Real; :: thesis: for Z being open Subset of REAL
for f1, f2, f being PartFunc of REAL ,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2 ) ) )
let Z be open Subset of REAL ; :: thesis: for f1, f2, f being PartFunc of REAL ,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2 ) ) )
let f1, f2, f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2 ) ) ) )
assume that
A1: Z c= dom (f1 + f2) and
A2: for x being Real st x in Z holds
f1 . x = 1 and
A3: ( f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) ) ; :: thesis: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2 ) ) )
A4: for x being Real st x in Z holds
f . x = ((1 / r) * x) + 0

proof

let x be Real; :: thesis: ( x in Z implies f . x = ((1 / r) * x) + 0 )
assume x in Z ; :: thesis: f . x = ((1 / r) * x) + 0
hence f . x = x / r by A3
.= ((1 / r) * x) + 0 ;
:: thesis: verum

end;

AA: Z c= (dom f1) /\ (dom f2) by A1, VALUED_1:def 1;
then A5: ( Z c= dom f1 & Z c= dom f2 ) by XBOOLE_1:18;
A6: for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A2;
then A7: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 0 ) ) by A5, FDIFF_1:31;
for x being Real st x in Z holds
f2 is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies f2 is_differentiable_in x )
assume A8: x in Z ; :: thesis: f2 is_differentiable_in x
Z c= dom ((#Z 2) * f) by A3, AA, XBOOLE_1:18;
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then Z c= dom f by TARSKI:def 3;
then f is_differentiable_on Z by A4, FDIFF_1:31;
then f is_differentiable_in x by A8, FDIFF_1:16;
hence f2 is_differentiable_in x by A3, TAYLOR_1:3; :: thesis: verum

end;

then A9: f2 is_differentiable_on Z by A5, FDIFF_1:16;
A10: for x being Real st x in Z holds
(f2 `| Z) . x = (2 * x) / (r ^2 )

proof

let x be Real; :: thesis: ( x in Z implies (f2 `| Z) . x = (2 * x) / (r ^2 ) )
assume A11: x in Z ; :: thesis: (f2 `| Z) . x = (2 * x) / (r ^2 )
Z c= dom ((#Z 2) * f) by A3, AA, XBOOLE_1:18;
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A12: Z c= dom f by TARSKI:def 3;
then A13: f is_differentiable_on Z by A4, FDIFF_1:31;
then A14: f is_differentiable_in x by A11, FDIFF_1:16;
(f2 `| Z) . x = diff ((#Z 2) * f),x by A3, A9, A11, FDIFF_1:def 8
.= (2 * ((f . x) #Z (2 - 1))) * (diff f,x) by A14, TAYLOR_1:3
.= (2 * (f . x)) * (diff f,x) by PREPOWER:45
.= (2 * (x / r)) * (diff f,x) by A3, A11
.= (2 * (x / r)) * ((f `| Z) . x) by A11, A13, FDIFF_1:def 8
.= (2 * (x / r)) * (1 / r) by A4, A11, A12, FDIFF_1:31
.= 2 * ((x / r) * (1 / r))
.= 2 * ((x * 1) / (r * r)) by XCMPLX_1:77
.= (2 * x) / (r ^2 ) ;
hence (f2 `| Z) . x = (2 * x) / (r ^2 ) ; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((f1 + f2) `| Z) . x = (2 * x) / (r ^2 ) )
assume A15: x in Z ; :: thesis: ((f1 + f2) `| Z) . x = (2 * x) / (r ^2 )
then ((f1 + f2) `| Z) . x = (diff f1,x) + (diff f2,x) by A1, A7, A9, FDIFF_1:26
.= ((f1 `| Z) . x) + (diff f2,x) by A7, A15, FDIFF_1:def 8
.= ((f1 `| Z) . x) + ((f2 `| Z) . x) by A9, A15, FDIFF_1:def 8
.= 0 + ((f2 `| Z) . x) by A5, A6, A15, FDIFF_1:31
.= (2 * x) / (r ^2 ) by A10, A15 ;
hence ((f1 + f2) `| Z) . x = (2 * x) / (r ^2 ) ; :: 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) / (r ^2 ) ) ) by A1, A7, A9, FDIFF_1:26; :: thesis: verum