let Z be open Subset of REAL; :: thesis:
assume A1: Z c= dom (sin * sec) ; :: thesis:
dom (sin * 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
sin * sec is_differentiable_in x

let x be Real; :: thesis:
assume x in Z ; :: thesis:
then cos . x <> 0 by A2, RFUNCT_1:3;
then A4: sec is_differentiable_in x by FDIFF_9:1;
sin is_differentiable_in sec . x by SIN_COS:64;
hence sin * sec is_differentiable_in x by A4, FDIFF_2:13; :: thesis:

end;

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

let x be Real; :: thesis:
A6: sin is_differentiable_in sec . x by SIN_COS:64;
assume A7: x in Z ; :: thesis:
then A8: cos . x <> 0 by A2, RFUNCT_1:3;
then sec is_differentiable_in x by FDIFF_9:1;
then diff ((sin * sec),x) = (diff (sin,(sec . x))) * (diff (sec,x)) by A6, FDIFF_2:13
.= (cos (sec . x)) * (diff (sec,x)) by SIN_COS:64
.= (cos (sec . x)) * ((sin . x) / ((cos . x) ^2)) by A8, FDIFF_9:1
.= ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2) ;
hence ((sin * sec) `| Z) . x = ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2) by A5, A7, FDIFF_1:def 7; :: thesis:

end;

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