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 A2: Z c= dom ((#R (1 / 2)) * f) by VALUED_1:8;
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A3: Z c= dom (f1 + ((- 1) (#) f2)) by A1, TARSKI:def 3;
A4: ( f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = (a ^2 ) + (0 * x) ) ) by A1;
then A5: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 + ((2 * (- 1)) * x) ) ) by A1, A3, Th12;

now

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * f is_differentiable_in x )
assume A6: x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x
then A7: f is_differentiable_in x by A5, FDIFF_1:16;
f . x > 0 by A1, A6;
hence (#R (1 / 2)) * f is_differentiable_in x by A7, TAYLOR_1:22; :: thesis: verum

end;

then A8: (#R (1 / 2)) * f is_differentiable_on Z by A2, FDIFF_1:16;
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 A9: x in Z ; :: thesis: (((- 1) (#) ((#R (1 / 2)) * f)) `| Z) . x = x * (((a ^2 ) - (x |^ 2)) #R (- (1 / 2)))
then A10: x in dom (f1 - f2) by A1, A2, FUNCT_1:21;
A11: f is_differentiable_in x by A5, A9, FDIFF_1:16;
A12: (f1 - f2) . x = (f1 . x) - (f2 . x) by A10, VALUED_1:13
.= (a ^2 ) - (f2 . x) by A1, A9
.= (a ^2 ) - (x #Z 2) by A1, TAYLOR_1:def 1
.= (a ^2 ) - (x |^ 2) by PREPOWER:46 ;
then A13: ( f . x = (a ^2 ) - (x |^ 2) & f . x > 0 ) by A1, A9;
(((- 1) (#) ((#R (1 / 2)) * f)) `| Z) . x = (- 1) * (diff ((#R (1 / 2)) * f),x) by A1, A8, A9, FDIFF_1:28
.= (- 1) * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff f,x)) by A11, A13, TAYLOR_1:22
.= (- 1) * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)) by A5, A9, FDIFF_1:def 8
.= (- 1) * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * x))) by A1, A3, A4, A9, Th12
.= x * (((a ^2 ) - (x |^ 2)) #R (- (1 / 2))) by A1, 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, A8, FDIFF_1:28; :: thesis: verum