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_1:44;
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_1:13;
then A4: sec is_differentiable_in x by FDIFF_9:1;
cos is_differentiable_in sec . x by SIN_COS:68;
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_1:16;
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 )) )
assume A6: x in Z ; :: thesis: ((cos * sec ) `| Z) . x = - (((sin . (sec . x)) * (sin . x)) / ((cos . x) ^2 ))
then A7: cos . x <> 0 by A2, RFUNCT_1:13;
then A8: sec is_differentiable_in x by FDIFF_9:1;
cos is_differentiable_in sec . x by SIN_COS:68;
then diff (cos * sec ),x = (diff cos ,(sec . x)) * (diff sec ,x) by A8, FDIFF_2:13
.= (- (sin (sec . x))) * (diff sec ,x) by SIN_COS:68
.= (- (sin (sec . x))) * ((sin . x) / ((cos . x) ^2 )) by A7, 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, A6, FDIFF_1:def 8; :: 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_1:16; :: thesis: verum