let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((cos / sin) * f) & ( for x being Real st x in Z holds
( f . x = x / 2 & sin . (f . x) <> 0 ) ) holds
( (cos / sin) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos / sin) * f) `| Z) . x = - (1 / (1 - (cos . x))) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((cos / sin) * f) & ( for x being Real st x in Z holds
( f . x = x / 2 & sin . (f . x) <> 0 ) ) implies ( (cos / sin) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos / sin) * f) `| Z) . x = - (1 / (1 - (cos . x))) ) ) )
assume that
A1: Z c= dom ((cos / sin) * f) and
A2: for x being Real st x in Z holds
( f . x = x / 2 & sin . (f . x) <> 0 ) ; :: thesis: ( (cos / sin) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos / sin) * f) `| Z) . x = - (1 / (1 - (cos . x))) ) )
A3: for x being Real st x in Z holds
f . x = ((1 / 2) * x) + 0

proof

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

end;

for y being object st y in Z holds
y in dom f by A1, FUNCT_1:11;
then A4: Z c= dom f by TARSKI:def 3;
then A5: f is_differentiable_on Z by A3, FDIFF_1:23;
A6: for x being Real st x in Z holds
(cos / sin) * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies (cos / sin) * f is_differentiable_in x )
assume A7: x in Z ; :: thesis: (cos / sin) * f is_differentiable_in x
then sin . (f . x) <> 0 by A2;
then A8: cos / sin is_differentiable_in f . x by Th47;
f is_differentiable_in x by A5, A7, FDIFF_1:9;
hence (cos / sin) * f is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

end;

then A9: (cos / sin) * f is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
(((cos / sin) * f) `| Z) . x = - (1 / (1 - (cos . x)))

proof

let x be Real; :: thesis: ( x in Z implies (((cos / sin) * f) `| Z) . x = - (1 / (1 - (cos . x))) )
assume A10: x in Z ; :: thesis: (((cos / sin) * f) `| Z) . x = - (1 / (1 - (cos . x)))
then A11: f is_differentiable_in x by A5, FDIFF_1:9;
A12: sin . (f . x) <> 0 by A2, A10;
then cos / sin is_differentiable_in f . x by Th47;
then diff (((cos / sin) * f),x) = (diff ((cos / sin),(f . x))) * (diff (f,x)) by A11, FDIFF_2:13
.= (- (1 / ((sin . (f . x)) ^2))) * (diff (f,x)) by A12, Th47
.= - ((1 / ((sin . (f . x)) ^2)) * (diff (f,x)))
.= - ((diff (f,x)) / ((sin . (f . x)) ^2)) by XCMPLX_1:99
.= - ((diff (f,x)) / ((sin . (x / 2)) ^2)) by A2, A10
.= - (((f `| Z) . x) / ((sin . (x / 2)) ^2)) by A5, A10, FDIFF_1:def 7
.= - ((1 / 2) / ((sin . (x / 2)) ^2)) by A3, A4, A10, FDIFF_1:23
.= - (1 / (2 * ((sin . (x / 2)) ^2))) by XCMPLX_1:78
.= - (1 / (1 - (cos . x))) by Lm2 ;
hence (((cos / sin) * f) `| Z) . x = - (1 / (1 - (cos . x))) by A9, A10, FDIFF_1:def 7; :: thesis: verum

end;

hence ( (cos / sin) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos / sin) * f) `| Z) . x = - (1 / (1 - (cos . x))) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum