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 #Z 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 #Z 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 #Z 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 #Z 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 + ((- 1) (#) f2)) by A1, TARSKI:def 3;
A3: ( f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = (a ^2 ) + (0 * x) ) ) by A1;
then A4: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 + ((2 * (- 1)) * x) ) ) by A1, A2, FDIFF_4:12;
A5: 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 A6: x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x
then A7: f is_differentiable_in x by A4, 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 A1, FDIFF_1:16;
for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = - (x * (((a ^2 ) - (x #Z 2)) #R (- (1 / 2))))

proof

let x be Real; :: thesis: ( x in Z implies (((#R (1 / 2)) * f) `| Z) . x = - (x * (((a ^2 ) - (x #Z 2)) #R (- (1 / 2)))) )
assume A9: x in Z ; :: thesis: (((#R (1 / 2)) * f) `| Z) . x = - (x * (((a ^2 ) - (x #Z 2)) #R (- (1 / 2))))
then A10: x in dom (f1 - f2) by A1, FUNCT_1:21;
A11: f is_differentiable_in x by A4, 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 ;
then A13: ( f . x = (a ^2 ) - (x #Z 2) & f . x > 0 ) by A1, A9;
(((#R (1 / 2)) * f) `| Z) . x = diff ((#R (1 / 2)) * f),x by A8, A9, FDIFF_1:def 8
.= ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff f,x) by A11, A13, TAYLOR_1:22
.= ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x) by A4, A9, FDIFF_1:def 8
.= ((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * x)) by A1, A2, A3, A9, FDIFF_4:12
.= - (x * (((a ^2 ) - (x #Z 2)) #R (- (1 / 2)))) by A1, A12 ;
hence (((#R (1 / 2)) * f) `| Z) . x = - (x * (((a ^2 ) - (x #Z 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 #Z 2)) #R (- (1 / 2)))) ) ) by A1, A5, FDIFF_1:16; :: thesis: verum