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: Z c= dom ((#R (1 / 2)) * f) by A1, VALUED_1:def 5;
then A6: for y being set st y in Z holds
y in dom f by FUNCT_1:21;
A7: f2 = 1 (#) f2 by RFUNCT_1:33;
then A8: Z c= dom (f1 + (1 (#) f2)) by A2, A6, TARSKI:def 3;
for x being Real st x in Z holds
f1 . x = 0 + (1 * x) by A4;
then A9: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 1 + ((2 * 1) * x) ) ) by A3, A7, A8, Th12;

now

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * f is_differentiable_in x )
assume A10: x in Z ; :: thesis: (#R (1 / 2)) * f is_differentiable_in x
then A11: f is_differentiable_in x by A2, A9, FDIFF_1:16;
f . x > 0 by A4, A10;
hence (#R (1 / 2)) * f is_differentiable_in x by A11, TAYLOR_1:22; :: thesis: verum

end;

then A12: (#R (1 / 2)) * f is_differentiable_on Z by A5, FDIFF_1:16;
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 A13: x in Z ; :: thesis: ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2)))
then A14: x in dom (f1 + f2) by A2, A5, FUNCT_1:21;
A15: f is_differentiable_in x by A2, A9, A13, FDIFF_1:16;
A16: (f1 + f2) . x = (f1 . x) + (f2 . x) by A14, VALUED_1:def 1
.= x + (f2 . x) by A4, A13
.= x + (x #Z 2) by A3, TAYLOR_1:def 1
.= x + (x |^ 2) by PREPOWER:46 ;
then A17: ( f . x = x + (x |^ 2) & f . x > 0 ) by A2, A4, A13;
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = 2 * (diff ((#R (1 / 2)) * f),x) by A1, A12, A13, FDIFF_1:28
.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * (diff f,x)) by A15, A17, TAYLOR_1:22
.= 2 * (((1 / 2) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)) by A2, A9, A13, FDIFF_1:def 8
.= ((2 * (1 / 2)) * ((f . x) #R ((1 / 2) - 1))) * ((f `| Z) . x)
.= ((2 * x) + 1) * (((x |^ 2) + x) #R (- (1 / 2))) by A2, A9, A13, A16 ;
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, A12, FDIFF_1:28; :: thesis: verum