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) ) )
A2: 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 A3: cos . x <> 0 by FDIFF_8:1;
( sin is_differentiable_in x & cos is_differentiable_in x ) by SIN_COS:63, SIN_COS:64;
hence tan is_differentiable_in x by A3, FDIFF_2:14; :: thesis: verum

end;

then A4: tan is_differentiable_on Z by A1, FDIFF_1:9;
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) )
A5: ( sin is_differentiable_in x & cos is_differentiable_in x ) by SIN_COS:63, SIN_COS:64;
assume A6: 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 A5, FDIFF_2:14
.= (((cos . x) * (cos . x)) - ((diff (cos,x)) * (sin . x))) / ((cos . x) ^2) by SIN_COS:64
.= (((cos . x) * (cos . x)) - ((- (sin . x)) * (sin . x))) / ((cos . x) ^2) by SIN_COS:63
.= (((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) / ((cos . x) ^2)
.= 1 / ((cos . x) ^2) by SIN_COS:28 ;
hence (tan `| Z) . x = 1 / ((cos . x) ^2) by A4, A6, FDIFF_1:def 7; :: 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, A2, FDIFF_1:9; :: thesis: verum