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 A1: ( 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 ) ) ) ; :: 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))) ) )
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A2: ( Z c= dom (f1 + f2) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = a ^2 ) ) by A1, TARSKI:def 3;
then A3: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 2 * x ) ) by A1, Th17;
A4: 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 A5: x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x
then A6: f is_differentiable_in x by A3, FDIFF_1:16;
f . x > 0 by A1, A5;
hence (#R (1 / 2)) * f is_differentiable_in x by A6, TAYLOR_1:22; :: thesis: verum

end;

then A7: (#R (1 / 2)) * f is_differentiable_on Z by A1, FDIFF_1:16;
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 A8: x in Z ; :: thesis: (((#R (1 / 2)) * f) `| Z) . x = x * (((a ^2 ) + (x |^ 2)) #R (- (1 / 2)))
then A9: x in dom (f1 + f2) by A1, FUNCT_1:21;
A10: f is_differentiable_in x by A3, A8, FDIFF_1:16;
A11: (f1 + f2) . x = (f1 . x) + (f2 . x) by A9, VALUED_1:def 1
.= (a ^2 ) + (f2 . x) by A1, A8
.= (a ^2 ) + (x #Z 2) by A1, TAYLOR_1:def 1
.= (a ^2 ) + (x |^ 2) by PREPOWER:46 ;
then ( f . x = (a ^2 ) + (x |^ 2) & f . x > 0 ) by A1, A8;
then diff ((#R (1 / 2)) * f),x = ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff f,x) by A10, TAYLOR_1:22
.= ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x) by A3, A8, FDIFF_1:def 8
.= ((1 / 2) * (((a ^2 ) + (x |^ 2)) #R ((1 / 2) - 1))) * (2 * x) by A1, A2, A8, 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 A7, A8, FDIFF_1:def 8; :: 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, A4, FDIFF_1:16; :: thesis: verum