let Z be open Subset of REAL; :: thesis: ( Z c= dom (cos * sec) implies ( cos * sec is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * sec) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) ) ) )
assume A1: Z c= dom (cos * sec) ; :: thesis: ( cos * sec is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * sec) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) ) )
dom (cos * sec) c= dom sec by RELAT_1:25;
then A2: Z c= dom sec by A1, XBOOLE_1:1;
A3: for x being Real st x in Z holds
cos * sec is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cos * sec is_differentiable_in x )
assume x in Z ; :: thesis: cos * sec is_differentiable_in x
then cos . x <> 0 by A2, RFUNCT_1:3;
then A4: sec is_differentiable_in x by FDIFF_9:1;
cos is_differentiable_in sec . x by SIN_COS:63;
hence cos * sec is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: cos * sec is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cos * sec) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((cos * sec) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) )
A6: cos is_differentiable_in sec . x by SIN_COS:63;
assume A7: x in Z ; :: thesis: ((cos * sec) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2))
then A8: cos . x <> 0 by A2, RFUNCT_1:3;
then sec is_differentiable_in x by FDIFF_9:1;
then diff ((cos * sec),x) = (diff (cos,(sec . x))) * (diff (sec,x)) by A6, FDIFF_2:13
.= (- (sin (sec . x))) * (diff (sec,x)) by SIN_COS:63
.= (- (sin (sec . x))) * ((sin . x) / ((cos . x) ^2)) by A8, FDIFF_9:1
.= - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) ;
hence ((cos * sec) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) by A5, A7, FDIFF_1:def 7; :: thesis: verum

end;

hence ( cos * sec is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * sec) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2)) ) ) by A1, A3, FDIFF_1:9; :: thesis: verum