let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (2 (#) ((#R (1 / 2)) * (f - cos))) & ( for x being Real st x in Z holds
( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) holds
( 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) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * (f - cos))) & ( for x being Real st x in Z holds
( f . x = 1 & sin . x > 0 & cos . x < 1 & cos . x > - 1 ) ) implies ( 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) ) ) )
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: ( 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) ) )
A3: for x being Real st x in Z holds
f . x = (0 * x) + 1 by A2;
A4: Z c= dom ((#R (1 / 2)) * (f - cos)) by A1, VALUED_1:def 5;
then for y being object st y in Z holds
y in dom (f - cos) by FUNCT_1:11;
then A5: Z c= dom (f - cos) by TARSKI:def 3;
then Z c= (dom cos) /\ (dom f) by VALUED_1:12;
then A6: Z c= dom f by XBOOLE_1:18;
then A7: f is_differentiable_on Z by A3, FDIFF_1:23;
A8: cos is_differentiable_on Z by FDIFF_1:26, SIN_COS:67;
then A9: f - cos is_differentiable_on Z by A5, A7, FDIFF_1:19;
A10: for x being Real st x in Z holds
(f - cos) . x > 0

proof

let x be Real; :: thesis: ( x in Z implies (f - cos) . x > 0 )
assume A11: x in Z ; :: thesis: (f - cos) . x > 0
then cos . x < 1 by A2;
then A12: 1 - (cos . x) > 1 - 1 by XREAL_1:15;
(f - cos) . x = (f . x) - (cos . x) by A5, A11, VALUED_1:13
.= 1 - (cos . x) by A2, A11 ;
hence (f - cos) . x > 0 by A12; :: thesis: verum

end;

now :: thesis: for x being Real st x in Z holds
(#R (1 / 2)) * (f - cos) is_differentiable_in x

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * (f - cos) is_differentiable_in x )
assume x in Z ; :: thesis: (#R (1 / 2)) * (f - cos) is_differentiable_in x
then ( f - cos is_differentiable_in x & (f - cos) . x > 0 ) by A9, A10, FDIFF_1:9;
hence (#R (1 / 2)) * (f - cos) is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

end;

then A13: (#R (1 / 2)) * (f - cos) is_differentiable_on Z by A4, FDIFF_1:9;
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: ( x in Z implies ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) )
assume A14: x in Z ; :: thesis: ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2)
then A15: diff ((f - cos),x) = ((f - cos) `| Z) . x by A9, FDIFF_1:def 7
.= (diff (f,x)) - (diff (cos,x)) by A5, A7, A8, A14, FDIFF_1:19
.= ((f `| Z) . x) - (diff (cos,x)) by A7, A14, FDIFF_1:def 7
.= 0 - (diff (cos,x)) by A6, A3, A14, FDIFF_1:23
.= 0 - (- (sin . x)) by SIN_COS:63
.= sin . x ;
A16: cos . x > - 1 by A2, A14;
A17: (f - cos) . x = (f . x) - (cos . x) by A5, A14, VALUED_1:13
.= 1 - (cos . x) by A2, A14 ;
A18: ( f - cos is_differentiable_in x & (f - cos) . x > 0 ) by A9, A10, A14, FDIFF_1:9;
A19: ( sin . x > 0 & cos . x < 1 ) by A2, A14;
((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = 2 * (diff (((#R (1 / 2)) * (f - cos)),x)) by A1, A13, A14, FDIFF_1:20
.= 2 * (((1 / 2) * (((f - cos) . x) #R ((1 / 2) - 1))) * (diff ((f - cos),x))) by A18, TAYLOR_1:22
.= (sin . x) * ((1 - (cos . x)) #R (- (1 / 2))) by A17, A15
.= (sin . x) * (1 / ((1 - (cos . x)) #R (1 / 2))) by A10, A14, A17, PREPOWER:76
.= (sin . x) / ((1 - (cos . x)) #R (1 / 2)) by XCMPLX_1:99
.= (1 + (cos . x)) #R (1 / 2) by A19, A16, Lm4 ;
hence ((2 (#) ((#R (1 / 2)) * (f - cos))) `| Z) . x = (1 + (cos . x)) #R (1 / 2) ; :: thesis: verum

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, A13, FDIFF_1:20; :: thesis: verum