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

proof

let x be Real; :: thesis: ( x in Z implies sin * sec is_differentiable_in x )
assume x in Z ; :: thesis: sin * 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;
sin is_differentiable_in sec . x by SIN_COS:69;
hence sin * sec is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: sin * sec is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sin * sec ) `| Z) . x = ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((sin * sec ) `| Z) . x = ((cos . (sec . x)) * (sin . x)) / ((cos . x) ^2 ) )
assume A6: x in Z ; :: thesis: ((sin * sec ) `| Z) . x = ((cos . (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;
sin is_differentiable_in sec . x by SIN_COS:69;
then diff (sin * sec ),x = (diff sin ,(sec . x)) * (diff sec ,x) by A8, FDIFF_2:13
.= (cos (sec . x)) * (diff sec ,x) by SIN_COS:69
.= (cos (sec . x)) * ((sin . x) / ((cos . x) ^2 )) by A7, 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, A6, FDIFF_1:def 8; :: thesis: verum

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_1:16; :: thesis: verum