let a be Real; :: thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((#R (1 / 2)) * f) & f = f1 + f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 ) ) holds
( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2))) ) )
let Z be open Subset of REAL; :: thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((#R (1 / 2)) * f) & f = f1 + f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 ) ) holds
( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2))) ) )
let f, f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((#R (1 / 2)) * f) & f = f1 + f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 ) ) implies ( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2))) ) ) )
assume that
A1: Z c= dom ((#R (1 / 2)) * f) and
A2: f = f1 + f2 and
A3: f2 = #Z 2 and
A4: for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 ) ; :: thesis: ( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2))) ) )
for y being object st y in Z holds
y in dom f by A1, FUNCT_1:11;
then A5: Z c= dom (f1 + f2) by A2, TARSKI:def 3;
A6: for x being Real st x in Z holds
f1 . x = a ^2 by A4;
then A7: f is_differentiable_on Z by A2, A3, A5, Th17;
A8: for x being Real st x in Z holds
(#R (1 / 2)) * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * f is_differentiable_in x )
assume x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x
then ( f is_differentiable_in x & f . x > 0 ) by A4, A7, FDIFF_1:9;
hence (#R (1 / 2)) * f is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

end;

then A9: (#R (1 / 2)) * f is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2)))

proof

let x be Real; :: thesis: ( x in Z implies (((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2))) )
assume A10: x in Z ; :: thesis: (((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2)))
then x in dom (f1 + f2) by A1, A2, FUNCT_1:11;
then A11: (f1 + f2) . x = (f1 . x) + (f2 . x) by VALUED_1:def 1
.= (a ^2) + (f2 . x) by A4, A10
.= (a ^2) + (x #Z 2) by A3, TAYLOR_1:def 1
.= (a ^2) + (x |^ 2) by PREPOWER:36 ;
( f is_differentiable_in x & f . x > 0 ) by A4, A7, A10, FDIFF_1:9;
then diff (((#R (1 / 2)) * f),x) = ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff (f,x)) by TAYLOR_1:22
.= ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x) by A7, A10, FDIFF_1:def 7
.= ((1 / 2) * (((a ^2) + (x |^ 2)) #R ((1 / 2) - 1))) * (2 * x) by A2, A3, A5, A6, A10, A11, Th17
.= x * (((a ^2) + (x |^ 2)) #R (- (1 / 2))) ;
hence (((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2))) by A9, A10, FDIFF_1:def 7; :: thesis: verum

end;

hence ( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2) + (x |^ 2)) #R (- (1 / 2))) ) ) by A1, A8, FDIFF_1:9; :: thesis: verum