let Z be open Subset of REAL; :: thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * f)) & f = f1 + f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = x & f . x > 0 ) ) holds
( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2))) ) )
let f, f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * f)) & f = f1 + f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = x & f . x > 0 ) ) implies ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2))) ) ) )
assume that
A1: Z c= dom (2 (#) ((#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 = x & f . x > 0 ) ; :: thesis: ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2))) ) )
A5: for x being Real st x in Z holds
f1 . x = 0 + (1 * x) by A4;
A6: f2 = 1 (#) f2 by RFUNCT_1:21;
A7: Z c= dom ((#R (1 / 2)) * f) by A1, VALUED_1:def 5;
then for y being object st y in Z holds
y in dom f by FUNCT_1:11;
then A8: Z c= dom (f1 + (1 (#) f2)) by A2, A6, TARSKI:def 3;
then A9: f1 + f2 is_differentiable_on Z by A3, A6, 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 A2, A4, A9, FDIFF_1:9;
hence (#R (1 / 2)) * f is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

end;

then A10: (#R (1 / 2)) * f is_differentiable_on Z by A7, FDIFF_1:9;
A11: for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 1 + ((2 * 1) * x) by A3, A6, A8, A5, Th12;
for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2)))

proof

let x be Real; :: thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2))) )
assume A12: x in Z ; :: thesis: ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2)))
then A13: ( f is_differentiable_in x & f . x > 0 ) by A2, A4, A9, FDIFF_1:9;
x in dom (f1 + f2) by A2, A7, A12, FUNCT_1:11;
then A14: (f1 + f2) . x = (f1 . x) + (f2 . x) by VALUED_1:def 1
.= x + (f2 . x) by A4, A12
.= x + (x #Z 2) by A3, TAYLOR_1:def 1
.= x + (x |^ 2) by PREPOWER:36 ;
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = 2 * (diff (((#R (1 / 2)) * f),x)) by A1, A10, A12, FDIFF_1:20
.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff (f,x))) by A13, TAYLOR_1:22
.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)) by A2, A9, A12, FDIFF_1:def 7
.= ((2 * (1 / 2)) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)
.= ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2))) by A2, A11, A12, A14 ;
hence ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2))) ; :: thesis: verum

end;

hence ( 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2))) ) ) by A1, A10, FDIFF_1:20; :: thesis: verum