let Z be open Subset of REAL ; :: thesis: ( not 0 in Z & Z c= dom (((id Z) ^ ) (#) tan ) implies ( ((id Z) ^ ) (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) tan ) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 )) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (((id Z) ^ ) (#) tan ) ; :: thesis: ( ((id Z) ^ ) (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) tan ) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 )) ) )
A3: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
A4: Z c= (dom ((id Z) ^ )) /\ (dom tan ) by A2, VALUED_1:def 4;
then A5: Z c= dom tan by XBOOLE_1:18;
A6: for x being Real st x in Z holds
( tan is_differentiable_in x & diff tan ,x = 1 / ((cos . x) ^2 ) )

proof

let x be Real; :: thesis: ( x in Z implies ( tan is_differentiable_in x & diff tan ,x = 1 / ((cos . x) ^2 ) ) )
assume x in Z ; :: thesis: ( tan is_differentiable_in x & diff tan ,x = 1 / ((cos . x) ^2 ) )
then cos . x <> 0 by A5, Th1;
hence ( tan is_differentiable_in x & diff tan ,x = 1 / ((cos . x) ^2 ) ) by FDIFF_7:46; :: thesis: verum

end;

then for x being Real st x in Z holds
tan is_differentiable_in x ;
then A7: tan is_differentiable_on Z by A5, FDIFF_1:16;
A8: Z c= dom ((id Z) ^ ) by A4, XBOOLE_1:18;
for x being Real st x in Z holds
((((id Z) ^ ) (#) tan ) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((((id Z) ^ ) (#) tan ) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 )) )
assume A9: x in Z ; :: thesis: ((((id Z) ^ ) (#) tan ) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 ))
then ((((id Z) ^ ) (#) tan ) `| Z) . x = ((tan . x) * (diff ((id Z) ^ ),x)) + ((((id Z) ^ ) . x) * (diff tan ,x)) by A2, A3, A7, FDIFF_1:29
.= ((tan . x) * ((((id Z) ^ ) `| Z) . x)) + ((((id Z) ^ ) . x) * (diff tan ,x)) by A3, A9, FDIFF_1:def 8
.= ((tan . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (diff tan ,x)) by A1, A9, FDIFF_5:4
.= (- ((tan . x) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (1 / ((cos . x) ^2 ))) by A6, A9
.= (- (((sin . x) / (cos . x)) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) / ((cos . x) ^2 )) by A5, A9, RFUNCT_1:def 4
.= (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((((id Z) . x) " ) / ((cos . x) ^2 )) by A8, A9, RFUNCT_1:def 8
.= (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 )) by A9, FUNCT_1:35 ;
hence ((((id Z) ^ ) (#) tan ) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 )) ; :: thesis: verum

end;

hence ( ((id Z) ^ ) (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^ ) (#) tan ) `| Z) . x = (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 )) ) ) by A2, A3, A7, FDIFF_1:29; :: thesis: verum