let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((#Z 2) * (sin / cos )) & ( for x being Real st x in Z holds
cos . x <> 0 ) implies ( (#Z 2) * (sin / cos ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * (sin / cos )) `| Z) . x = (2 * (sin . x)) / ((cos . x) #Z 3) ) ) )
assume that
A1: Z c= dom ((#Z 2) * (sin / cos )) and
A2: for x being Real st x in Z holds
cos . x <> 0 ; :: thesis: ( (#Z 2) * (sin / cos ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * (sin / cos )) `| Z) . x = (2 * (sin . x)) / ((cos . x) #Z 3) ) )
for y being set st y in Z holds
y in dom (sin / cos ) by A1, FUNCT_1:21;
then A3: Z c= dom (sin / cos ) by TARSKI:def 3;
A4: for x being Real st x in Z holds
(#Z 2) * (sin / cos ) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies (#Z 2) * (sin / cos ) is_differentiable_in x )
assume x in Z ; :: thesis: (#Z 2) * (sin / cos ) is_differentiable_in x
then cos . x <> 0 by A2;
then sin / cos is_differentiable_in x by Th46;
hence (#Z 2) * (sin / cos ) is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

then A5: (#Z 2) * (sin / cos ) is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
(((#Z 2) * (sin / cos )) `| Z) . x = (2 * (sin . x)) / ((cos . x) #Z 3)

proof

let x be Real; :: thesis: ( x in Z implies (((#Z 2) * (sin / cos )) `| Z) . x = (2 * (sin . x)) / ((cos . x) #Z 3) )
assume A6: x in Z ; :: thesis: (((#Z 2) * (sin / cos )) `| Z) . x = (2 * (sin . x)) / ((cos . x) #Z 3)
then A7: cos . x <> 0 by A2;
then A8: sin / cos is_differentiable_in x by Th46;
A9: (sin / cos ) . x = (sin . x) * ((cos . x) " ) by A3, A6, RFUNCT_1:def 4
.= (sin . x) * (1 / (cos . x)) by XCMPLX_1:217
.= (sin . x) / (cos . x) by XCMPLX_1:100 ;
(((#Z 2) * (sin / cos )) `| Z) . x = diff ((#Z 2) * (sin / cos )),x by A5, A6, FDIFF_1:def 8
.= (2 * (((sin / cos ) . x) #Z (2 - 1))) * (diff (sin / cos ),x) by A8, TAYLOR_1:3
.= (2 * (((sin / cos ) . x) #Z (2 - 1))) * (1 / ((cos . x) ^2 )) by A7, Th46
.= (2 * (((sin / cos ) . x) #Z 1)) / ((cos . x) ^2 ) by XCMPLX_1:100
.= (2 * ((sin . x) / (cos . x))) / ((cos . x) ^2 ) by A9, PREPOWER:45
.= ((2 * (sin . x)) / (cos . x)) / ((cos . x) ^2 ) by XCMPLX_1:75
.= (2 * (sin . x)) / ((cos . x) * ((cos . x) ^2 )) by XCMPLX_1:79
.= (2 * (sin . x)) / ((cos . x) * ((cos . x) #Z 2)) by Th1
.= (2 * (sin . x)) / (((cos . x) #Z 1) * ((cos . x) #Z 2)) by PREPOWER:45
.= (2 * (sin . x)) / ((cos . x) #Z (1 + 2)) by A2, A6, PREPOWER:54
.= (2 * (sin . x)) / ((cos . x) #Z 3) ;
hence (((#Z 2) * (sin / cos )) `| Z) . x = (2 * (sin . x)) / ((cos . x) #Z 3) ; :: thesis: verum

end;

hence ( (#Z 2) * (sin / cos ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * (sin / cos )) `| Z) . x = (2 * (sin . x)) / ((cos . x) #Z 3) ) ) by A1, A4, FDIFF_1:16; :: thesis: verum