let Z be open Subset of REAL; :: thesis: ( Z c= dom (cos * cosec) implies ( cos * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * cosec) `| Z) . x = ((sin . (cosec . x)) * (cos . x)) / ((sin . x) ^2) ) ) )
assume A1: Z c= dom (cos * cosec) ; :: thesis: ( cos * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * cosec) `| Z) . x = ((sin . (cosec . x)) * (cos . x)) / ((sin . x) ^2) ) )
dom (cos * cosec) c= dom cosec by RELAT_1:25;
then A2: Z c= dom cosec by A1, XBOOLE_1:1;
A3: for x being Real st x in Z holds
cos * cosec is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cos * cosec is_differentiable_in x )
assume x in Z ; :: thesis: cos * cosec is_differentiable_in x
then sin . x <> 0 by A2, RFUNCT_1:3;
then A4: cosec is_differentiable_in x by FDIFF_9:2;
cos is_differentiable_in cosec . x by SIN_COS:63;
hence cos * cosec is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((cos * cosec) `| Z) . x = ((sin . (cosec . x)) * (cos . x)) / ((sin . x) ^2) )
A6: cos is_differentiable_in cosec . x by SIN_COS:63;
assume A7: x in Z ; :: thesis: ((cos * cosec) `| Z) . x = ((sin . (cosec . x)) * (cos . x)) / ((sin . x) ^2)
then A8: sin . x <> 0 by A2, RFUNCT_1:3;
then cosec is_differentiable_in x by FDIFF_9:2;
then diff ((cos * cosec),x) = (diff (cos,(cosec . x))) * (diff (cosec,x)) by A6, FDIFF_2:13
.= (- (sin (cosec . x))) * (diff (cosec,x)) by SIN_COS:63
.= (- (sin (cosec . x))) * (- ((cos . x) / ((sin . x) ^2))) by A8, FDIFF_9:2 ;
hence ((cos * cosec) `| Z) . x = ((sin . (cosec . x)) * (cos . x)) / ((sin . x) ^2) by A5, A7, FDIFF_1:def 7; :: thesis: verum

end;

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