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_1:21;
hence cos . (cos . x) <> 0 by RFUNCT_1:13; :: 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 A4: x in Z ; :: thesis: sec * cos is_differentiable_in x
A5: cos is_differentiable_in x by SIN_COS:68;
cos . (cos . x) <> 0 by A2, A4;
then sec is_differentiable_in cos . x by Th1;
hence sec * cos is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: sec * cos is_differentiable_on Z by A1, FDIFF_1:16;
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 A7: x in Z ; :: thesis: ((sec * cos ) `| Z) . x = - (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2 ))
A8: cos is_differentiable_in x by SIN_COS:68;
A9: cos . (cos . x) <> 0 by A2, A7;
then sec is_differentiable_in cos . x by Th1;
then diff (sec * cos ),x = (diff sec ,(cos . x)) * (diff cos ,x) by A8, FDIFF_2:13
.= ((sin . (cos . x)) / ((cos . (cos . x)) ^2 )) * (diff cos ,x) by A9, Th1
.= (- (sin . x)) * ((sin . (cos . x)) / ((cos . (cos . x)) ^2 )) by SIN_COS:68 ;
hence ((sec * cos ) `| Z) . x = - (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2 )) by A6, A7, FDIFF_1:def 8; :: 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_1:16; :: thesis: verum