let Z be open Subset of REAL; :: thesis: ( Z c= dom (sec * cos) implies ( sec * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * cos) `| Z) . x = - (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2)) ) ) )
assume A1: Z c= dom (sec * cos) ; :: thesis: ( sec * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * cos) `| Z) . x = - (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2)) ) )
A2: for x being Real st x in Z holds
cos . (cos . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies cos . (cos . x) <> 0 )
assume x in Z ; :: thesis: cos . (cos . x) <> 0
then cos . x in dom sec by A1, FUNCT_1:11;
hence cos . (cos . x) <> 0 by RFUNCT_1:3; :: thesis: verum

end;

A3: for x being Real st x in Z holds
sec * cos is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sec * cos is_differentiable_in x )
assume x in Z ; :: thesis: sec * cos is_differentiable_in x
then cos . (cos . x) <> 0 by A2;
then ( cos is_differentiable_in x & sec is_differentiable_in cos . x ) by Th1, SIN_COS:63;
hence sec * cos is_differentiable_in x by FDIFF_2:13; :: thesis: verum

end;

then A4: sec * cos is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sec * cos) `| Z) . x = - (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((sec * cos) `| Z) . x = - (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2)) )
assume A5: x in Z ; :: thesis: ((sec * cos) `| Z) . x = - (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2))
then A6: cos . (cos . x) <> 0 by A2;
then ( cos is_differentiable_in x & sec is_differentiable_in cos . x ) by Th1, SIN_COS:63;
then diff ((sec * cos),x) = (diff (sec,(cos . x))) * (diff (cos,x)) by FDIFF_2:13
.= ((sin . (cos . x)) / ((cos . (cos . x)) ^2)) * (diff (cos,x)) by A6, Th1
.= (- (sin . x)) * ((sin . (cos . x)) / ((cos . (cos . x)) ^2)) by SIN_COS:63 ;
hence ((sec * cos) `| Z) . x = - (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2)) by A4, A5, FDIFF_1:def 7; :: thesis: verum

end;

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