let Z be open Subset of REAL ; :: thesis:
let f be PartFunc of REAL ,REAL ; :: thesis:
assume that
A1: Z c= dom (2 (#) ((#R (1 / 2)) * (f - cos ))) and
A2: for x being Real st x in Z holds
( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ; :: thesis:
A3: Z c= dom ((#R (1 / 2)) * (f - cos )) by A1, VALUED_1:def 5;
then for y being set st y in Z holds
y in dom (f - cos ) by FUNCT_1:21;
then A4: Z c= dom (f - cos ) by TARSKI:def 3;
then Z c= (dom cos ) /\ (dom f) by VALUED_1:12;
then A5: Z c= dom f by XBOOLE_1:18;
A6: for x being Real st x in Z holds
f . x = (0 * x) + 1 by A2;
then A7: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 0 ) ) by A5, FDIFF_1:31;
A8: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;
then A9: f - cos is_differentiable_on Z by A4, A7, FDIFF_1:27;
A10: for x being Real st x in Z holds
(f - cos ) . x > 0

proof

let x be Real; :: thesis:
assume A11: x in Z ; :: thesis:
then A12: (f - cos ) . x = (f . x) - (cos . x) by A4, VALUED_1:13
.= 1 - (cos . x) by A2, A11 ;
cos . x < 1 by A2, A11;
then 1 - (cos . x) > 1 - 1 by XREAL_1:17;
hence (f - cos ) . x > 0 by A12; :: thesis:

end;

now

let x be Real; :: thesis:
assume A13: x in Z ; :: thesis:
then A14: f - cos is_differentiable_in x by A9, FDIFF_1:16;
(f - cos ) . x > 0 by A10, A13;
hence (#R (1 / 2)) * (f - cos ) is_differentiable_in x by A14, TAYLOR_1:22; :: thesis:

end;

then A15: (#R (1 / 2)) * (f - cos ) is_differentiable_on Z by A3, FDIFF_1:16;
for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * (f - cos ))) `| Z) . x = (1 + (cos . x)) #R (1 / 2)

proof

let x be Real; :: thesis:
assume A16: x in Z ; :: thesis:
then A17: f - cos is_differentiable_in x by A9, FDIFF_1:16;
A18: (f - cos ) . x > 0 by A10, A16;
A19: (f - cos ) . x = (f . x) - (cos . x) by A4, A16, VALUED_1:13
.= 1 - (cos . x) by A2, A16 ;
A20: diff (f - cos ),x = ((f - cos ) `| Z) . x by A9, A16, FDIFF_1:def 8
.= (diff f,x) - (diff cos ,x) by A4, A7, A8, A16, FDIFF_1:27
.= ((f `| Z) . x) - (diff cos ,x) by A7, A16, FDIFF_1:def 8
.= 0 - (diff cos ,x) by A5, A6, A16, FDIFF_1:31
.= 0 - (- (sin . x)) by SIN_COS:68
.= sin . x ;
A21: ( sin . x > 0 & cos . x < 1 & cos . x > - 1 ) by A2, A16;
((2 (#) ((#R (1 / 2)) * (f - cos ))) `| Z) . x = 2 * (diff ((#R (1 / 2)) * (f - cos )),x) by A1, A15, A16, FDIFF_1:28
.= 2 * (((1 / 2) * (((f - cos ) . x) #R ((1 / 2) - 1))) * (diff (f - cos ),x)) by A17, A18, TAYLOR_1:22
.= (sin . x) * ((1 - (cos . x)) #R (- (1 / 2))) by A19, A20
.= (sin . x) * (1 / ((1 - (cos . x)) #R (1 / 2))) by A10, A16, A19, PREPOWER:90
.= (sin . x) / ((1 - (cos . x)) #R (1 / 2)) by XCMPLX_1:100
.= (1 + (cos . x)) #R (1 / 2) by A21, Lm4 ;
hence ((2 (#) ((#R (1 / 2)) * (f - cos ))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ; :: thesis:

end;

hence ( 2 (#) ((#R (1 / 2)) * (f - cos )) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * (f - cos ))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ) ) by A1, A15, FDIFF_1:28; :: thesis: