let a be Real; :: thesis: for Z being open Subset of REAL
for f, f1 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 f, f1 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 f, f1 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 ) and
A3: 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 ) )
A4: Z c= (dom f1) /\ (dom ((1 / (4 * a)) (#) (sin * f))) by A1, VALUED_1:def 1;
then A5: Z c= dom ((1 / (4 * a)) (#) (sin * f)) by XBOOLE_1:18;
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;

A7: for x being Real st x in Z holds
f . x = (2 * a) * x by A2;
then A8: (1 / (4 * a)) (#) (sin * f) is_differentiable_on Z by A3, A5, Th43;
A9: Z c= dom f1 by A4, XBOOLE_1:18;
then A10: f1 is_differentiable_on Z by A6, FDIFF_1:23;
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 A11: 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, A8, A10, FDIFF_1:18
.= ((f1 `| Z) . x) + (diff (((1 / (4 * a)) (#) (sin * f)),x)) by A10, A11, FDIFF_1:def 7
.= ((f1 `| Z) . x) + ((((1 / (4 * a)) (#) (sin * f)) `| Z) . x) by A8, A11, FDIFF_1:def 7
.= ((f1 `| Z) . x) + ((1 / 2) * (cos ((2 * a) * x))) by A3, A7, A5, A11, Th43
.= (1 / 2) + ((1 / 2) * (cos ((2 * a) * x))) by A9, A6, A11, FDIFF_1:23
.= (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, A8, A10, FDIFF_1:18; :: thesis: verum