let Z be open Subset of REAL; :: thesis: ( Z c= dom (sec * ((id Z) ^)) implies ( - (sec * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (sec * ((id Z) ^))) `| Z) . x = (sin . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2)) ) ) )
assume A1: Z c= dom (sec * ((id Z) ^)) ; :: thesis: ( - (sec * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (sec * ((id Z) ^))) `| Z) . x = (sin . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2)) ) )
then A2: Z c= dom (- (sec * ((id Z) ^))) by VALUED_1:def 5;
A3: Z c= dom ((id Z) ^) by A1, FUNCT_1:101;
A4: not 0 in Z

proof

assume A5: 0 in Z ; :: thesis: contradiction
dom ((id Z) ^) = (dom (id Z)) \ ((id Z) " {0}) by RFUNCT_1:def 2
.= (dom (id Z)) \ {0} by Lm1, A5 ;
then not 0 in {0} by A5, A3, XBOOLE_0:def 5;
hence contradiction by TARSKI:def 1; :: thesis: verum

end;

then A6: sec * ((id Z) ^) is_differentiable_on Z by A1, FDIFF_9:8;
then A7: (- 1) (#) (sec * ((id Z) ^)) is_differentiable_on Z by A2, FDIFF_1:20;
for x being Real st x in Z holds
((- (sec * ((id Z) ^))) `| Z) . x = (sin . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((- (sec * ((id Z) ^))) `| Z) . x = (sin . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2)) )
assume A8: x in Z ; :: thesis: ((- (sec * ((id Z) ^))) `| Z) . x = (sin . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2))
((- (sec * ((id Z) ^))) `| Z) . x = ((- 1) (#) ((sec * ((id Z) ^)) `| Z)) . x by A6, FDIFF_2:19
.= (- 1) * (((sec * ((id Z) ^)) `| Z) . x) by VALUED_1:6
.= (- 1) * (- ((sin . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2)))) by A1, A4, A8, FDIFF_9:8
.= (sin . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2)) ;
hence ((- (sec * ((id Z) ^))) `| Z) . x = (sin . (1 / x)) / ((x ^2) * ((cos . (1 / x)) ^2)) ; :: thesis: verum

end;

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