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 A1: ( not 0 in Z & 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 )) ) )
A2: for x being Real st x in Z holds
(id Z) . x = x by FUNCT_1:35;
A3: Z c= (dom ((id Z) ^ )) /\ (dom tan ) by A1, VALUED_1:def 4;
then A4: Z c= dom ((id Z) ^ ) by XBOOLE_1:18;
A5: Z c= dom tan by A3, XBOOLE_1:18;
A8: ( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by FDIFF_5:4, A1;
A9: 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 A10: tan is_differentiable_on Z by A5, FDIFF_1:16;
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 A11: 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 A1, A8, A10, FDIFF_1:29
.= ((tan . x) * ((((id Z) ^ ) `| Z) . x)) + ((((id Z) ^ ) . x) * (diff tan ,x)) by A8, A11, FDIFF_1:def 8
.= ((tan . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (diff tan ,x)) by A11, FDIFF_5:4, A1
.= (- ((tan . x) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (1 / ((cos . x) ^2 ))) by A9, A11
.= (- (((sin . x) / (cos . x)) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) / ((cos . x) ^2 )) by A5, A11, RFUNCT_1:def 4
.= (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((((id Z) . x) " ) / ((cos . x) ^2 )) by A4, A11, RFUNCT_1:def 8
.= (- (((sin . x) / (cos . x)) / (x ^2 ))) + ((1 / x) / ((cos . x) ^2 )) by A2, A11 ;
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 A1, A8, A10, FDIFF_1:29; :: thesis: verum