let Z be open Subset of REAL ; :: thesis: ( Z c= dom (tan (#) sec ) implies ( tan (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) sec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2 )) ) ) )
assume A1: Z c= dom (tan (#) sec ) ; :: thesis: ( tan (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) sec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2 )) ) )
then A2: Z c= (dom tan ) /\ (dom sec ) by VALUED_1:def 4;
then A3: Z c= dom tan by XBOOLE_1:18;
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 cos . x <> 0 by A3, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A4: tan is_differentiable_on Z by A3, FDIFF_1:16;
A5: Z c= dom sec by A2, XBOOLE_1:18;
A6: for x being Real st x in Z holds
( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) )

proof

let x be Real; :: thesis: ( x in Z implies ( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) ) )
assume x in Z ; :: thesis: ( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) )
then cos . x <> 0 by A5, RFUNCT_1:13;
hence ( sec is_differentiable_in x & diff sec ,x = (sin . x) / ((cos . x) ^2 ) ) by Th1; :: thesis: verum

end;

then for x being Real st x in Z holds
sec is_differentiable_in x ;
then A7: sec is_differentiable_on Z by A5, FDIFF_1:16;
for x being Real st x in Z holds
((tan (#) sec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((tan (#) sec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2 )) )
assume A8: x in Z ; :: thesis: ((tan (#) sec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2 ))
then A9: cos . x <> 0 by A3, FDIFF_8:1;
((tan (#) sec ) `| Z) . x = ((sec . x) * (diff tan ,x)) + ((tan . x) * (diff sec ,x)) by A1, A4, A7, A8, FDIFF_1:29
.= ((sec . x) * (1 / ((cos . x) ^2 ))) + ((tan . x) * (diff sec ,x)) by A9, FDIFF_7:46
.= ((sec . x) * (1 / ((cos . x) ^2 ))) + ((tan . x) * ((sin . x) / ((cos . x) ^2 ))) by A6, A8
.= ((1 / ((cos . x) ^2 )) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2 )) by A5, A8, RFUNCT_1:def 8 ;
hence ((tan (#) sec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2 )) ; :: thesis: verum

end;

hence ( tan (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) sec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (cos . x)) + (((tan . x) * (sin . x)) / ((cos . x) ^2 )) ) ) by A1, A4, A7, FDIFF_1:29; :: thesis: verum