let Z be open Subset of REAL; :: thesis: ( not 0 in Z & Z c= dom (((id Z) ^) (#) sec) implies ( ((id Z) ^) (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) sec) `| Z) . x = (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2)) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (((id Z) ^) (#) sec) ; :: thesis: ( ((id Z) ^) (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) sec) `| Z) . x = (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2)) ) )
A3: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
A4: Z c= (dom ((id Z) ^)) /\ (dom sec) by A2, VALUED_1:def 4;
then A5: Z c= dom sec by 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_1:3;
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_1:9;
A8: Z c= dom ((id Z) ^) by A4, XBOOLE_1:18;
for x being Real st x in Z holds
((((id Z) ^) (#) sec) `| Z) . x = (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((((id Z) ^) (#) sec) `| Z) . x = (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2)) )
assume A9: x in Z ; :: thesis: ((((id Z) ^) (#) sec) `| Z) . x = (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2))
then ((((id Z) ^) (#) sec) `| Z) . x = ((sec . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (sec,x))) by A2, A3, A7, FDIFF_1:21
.= ((sec . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (sec,x))) by A3, A9, FDIFF_1:def 7
.= ((sec . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (sec,x))) by A1, A9, FDIFF_5:4
.= (- ((sec . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((sin . x) / ((cos . x) ^2))) by A6, A9
.= (- (((cos . x) ") * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((sin . x) / ((cos . x) ^2))) by A5, A9, RFUNCT_1:def 2
.= (- ((1 / (cos . x)) / (x ^2))) + ((((id Z) . x) ") * ((sin . x) / ((cos . x) ^2))) by A8, A9, RFUNCT_1:def 2
.= (- ((1 / (cos . x)) / (x ^2))) + ((1 / x) * ((sin . x) / ((cos . x) ^2))) by A9, FUNCT_1:18
.= (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2)) ;
hence ((((id Z) ^) (#) sec) `| Z) . x = (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2)) ; :: thesis: verum

end;

hence ( ((id Z) ^) (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) sec) `| Z) . x = (- ((1 / (cos . x)) / (x ^2))) + (((sin . x) / x) / ((cos . x) ^2)) ) ) by A2, A3, A7, FDIFF_1:21; :: thesis: verum