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))) ) )
A5: for x being Real st x in Z holds
f1 . x = (a ^2) + (0 * x) by A4;
A6: Z c= dom ((#R (1 / 2)) * f) by A1, VALUED_1:8;
then for y being object st y in Z holds
y in dom f by FUNCT_1:11;
then A7: Z c= dom (f1 + ((- 1) (#) f2)) by A2, TARSKI:def 3;
then A8: f is_differentiable_on Z by A2, A3, A5, Th12;

now :: thesis: for x being Real st x in Z holds
(#R (1 / 2)) * f is_differentiable_in x

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, A8, 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 A6, FDIFF_1:9;
for x being Real st x in Z holds
(((- 1) (#) ((#R (1 / 2)) * f)) `| Z) . x = x * (((a ^2) - (x |^ 2)) #R (- (1 / 2)))

proof

let x be Real; :: thesis: ( x in Z implies (((- 1) (#) ((#R (1 / 2)) * f)) `| Z) . x = x * (((a ^2) - (x |^ 2)) #R (- (1 / 2))) )
assume A10: x in Z ; :: thesis: (((- 1) (#) ((#R (1 / 2)) * f)) `| Z) . x = x * (((a ^2) - (x |^ 2)) #R (- (1 / 2)))
then A11: ( f is_differentiable_in x & f . x > 0 ) by A4, A8, FDIFF_1:9;
x in dom (f1 - f2) by A2, A6, A10, FUNCT_1:11;
then A12: (f1 - f2) . x = (f1 . x) - (f2 . x) by VALUED_1:13
.= (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 ;
(((- 1) (#) ((#R (1 / 2)) * f)) `| Z) . x = (- 1) * (diff (((#R (1 / 2)) * f),x)) by A1, A9, A10, FDIFF_1:20
.= (- 1) * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff (f,x))) by A11, TAYLOR_1:22
.= (- 1) * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)) by A8, A10, FDIFF_1:def 7
.= (- 1) * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * x))) by A2, A3, A7, A5, A10, Th12
.= x * (((a ^2) - (x |^ 2)) #R (- (1 / 2))) by A2, A12 ;
hence (((- 1) (#) ((#R (1 / 2)) * f)) `| Z) . x = x * (((a ^2) - (x |^ 2)) #R (- (1 / 2))) ; :: 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, A9, FDIFF_1:20; :: thesis: verum