let Z be open Subset of REAL ; :: thesis: ( Z c= dom tan implies ( tan is_differentiable_on Z & ( for x being Real st x in Z holds
(tan `| Z) . x = 1 / ((cos . x) ^2 ) ) ) )
assume A1: Z c= dom tan ; :: thesis: ( tan is_differentiable_on Z & ( for x being Real st x in Z holds
(tan `| Z) . x = 1 / ((cos . x) ^2 ) ) )
A3: for x being Real st x in Z holds
tan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan is_differentiable_in x )
assume x in Z ; :: thesis: tan is_differentiable_in x
then x in dom tan by A1;
then A4: cos . x <> 0 by FDIFF_8:1;
( sin is_differentiable_in x & cos is_differentiable_in x ) by SIN_COS:68, SIN_COS:69;
hence tan is_differentiable_in x by A4, FDIFF_2:14; :: thesis: verum

end;

then A5: tan is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
(tan `| Z) . x = 1 / ((cos . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies (tan `| Z) . x = 1 / ((cos . x) ^2 ) )
A6: ( sin is_differentiable_in x & cos is_differentiable_in x ) by SIN_COS:68, SIN_COS:69;
assume A7: x in Z ; :: thesis: (tan `| Z) . x = 1 / ((cos . x) ^2 )
then x in dom tan by A1;
then cos . x <> 0 by FDIFF_8:1;
then diff tan ,x = (((diff sin ,x) * (cos . x)) - ((diff cos ,x) * (sin . x))) / ((cos . x) ^2 ) by A6, FDIFF_2:14
.= (((cos . x) * (cos . x)) - ((diff cos ,x) * (sin . x))) / ((cos . x) ^2 ) by SIN_COS:69
.= (((cos . x) * (cos . x)) - ((- (sin . x)) * (sin . x))) / ((cos . x) ^2 ) by SIN_COS:68
.= (((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) / ((cos . x) ^2 )
.= 1 / ((cos . x) ^2 ) by SIN_COS:31 ;
hence (tan `| Z) . x = 1 / ((cos . x) ^2 ) by A5, A7, FDIFF_1:def 8; :: thesis: verum

end;

hence ( tan is_differentiable_on Z & ( for x being Real st x in Z holds
(tan `| Z) . x = 1 / ((cos . x) ^2 ) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum