let a be Real; :: thesis: for Z being open Subset of REAL
for f1, f being PartFunc of REAL ,REAL st Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds
( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) )
let Z be open Subset of REAL ; :: thesis: for f1, f being PartFunc of REAL ,REAL st Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds
( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) )
let f1, f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 implies ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) ) )
assume that
A1: Z c= dom (f1 + ((1 / (4 * a)) (#) (sin * f))) and
A2: ( ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 ) ; :: thesis: ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) )
A3: ( ( for x being Real st x in Z holds
f . x = (2 * a) * x ) & a <> 0 ) by A2;
Z c= (dom f1) /\ (dom ((1 / (4 * a)) (#) (sin * f))) by A1, VALUED_1:def 1;
then A4: ( Z c= dom ((1 / (4 * a)) (#) (sin * f)) & Z c= dom f1 ) by XBOOLE_1:18;
then A5: ( (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * a)) (#) (sin * f)) `| Z) . x = (1 / 2) * (cos ((2 * a) * x)) ) ) by A3, Th43;
A6: for x being Real st x in Z holds
f1 . x = ((1 / 2) * x) + 0

proof

let x be Real; :: thesis: ( x in Z implies f1 . x = ((1 / 2) * x) + 0 )
assume x in Z ; :: thesis: f1 . x = ((1 / 2) * x) + 0
then f1 . x = x / 2 by A2
.= ((1 / 2) * x) + 0 ;
hence f1 . x = ((1 / 2) * x) + 0 ; :: thesis: verum

end;

then A7: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 1 / 2 ) ) by A4, FDIFF_1:31;
for x being Real st x in Z holds
((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2

proof

let x be Real; :: thesis: ( x in Z implies ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 )
assume A8: x in Z ; :: thesis: ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2
then ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (diff f1,x) + (diff ((1 / (4 * a)) (#) (sin * f)),x) by A1, A5, A7, FDIFF_1:26
.= ((f1 `| Z) . x) + (diff ((1 / (4 * a)) (#) (sin * f)),x) by A7, A8, FDIFF_1:def 8
.= ((f1 `| Z) . x) + ((((1 / (4 * a)) (#) (sin * f)) `| Z) . x) by A5, A8, FDIFF_1:def 8
.= ((f1 `| Z) . x) + ((1 / 2) * (cos ((2 * a) * x))) by A3, A4, A8, Th43
.= (1 / 2) + ((1 / 2) * (cos ((2 * a) * x))) by A4, A6, A8, FDIFF_1:31
.= (1 / 2) * (1 + (cos (2 * (a * x))))
.= (1 / 2) * (2 * ((cos (a * x)) ^2 )) by Lm2
.= (cos (a * x)) ^2 ;
hence ((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ; :: thesis: verum

end;

hence ( f1 + ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (cos (a * x)) ^2 ) ) by A1, A5, A7, FDIFF_1:26; :: thesis: verum