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_1:171;
A4: not 0 in Z

proof

assume K: 0 in Z ; :: thesis: contradiction
dom ((id Z) ^ ) = (dom (id Z)) \ ((id Z) " {0 }) by RFUNCT_1:def 8
.= (dom (id Z)) \ {0 } by Lm0, K ;
then not 0 in {0 } by XBOOLE_0:def 5, K, A3;
hence contradiction by TARSKI:def 1; :: thesis: verum

end;

then A5: sec * ((id Z) ^ ) is_differentiable_on Z by A1, FDIFF_9:8;
then A6: (- 1) (#) (sec * ((id Z) ^ )) is_differentiable_on Z by A2, FDIFF_1:28, A;
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 A7: 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 A5, FDIFF_2:19, A
.= (- 1) * (((sec * ((id Z) ^ )) `| Z) . x) by VALUED_1:6
.= (- 1) * (- ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 )))) by A1, A4, A7, 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 A6; :: thesis: verum