let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cosec * cos ) implies ( cosec * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cos ) `| Z) . x = ((sin . x) * (cos . (cos . x))) / ((sin . (cos . x)) ^2 ) ) ) )
assume A1: Z c= dom (cosec * cos ) ; :: thesis: ( cosec * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cos ) `| Z) . x = ((sin . x) * (cos . (cos . x))) / ((sin . (cos . x)) ^2 ) ) )
A2: for x being Real st x in Z holds
sin . (cos . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . (cos . x) <> 0 )
assume x in Z ; :: thesis: sin . (cos . x) <> 0
then cos . x in dom cosec by A1, FUNCT_1:21;
hence sin . (cos . x) <> 0 by RFUNCT_1:13; :: thesis: verum

end;

A3: for x being Real st x in Z holds
cosec * cos is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cosec * cos is_differentiable_in x )
assume A4: x in Z ; :: thesis: cosec * cos is_differentiable_in x
A5: cos is_differentiable_in x by SIN_COS:68;
sin . (cos . x) <> 0 by A2, A4;
then cosec is_differentiable_in cos . x by Th2;
hence cosec * cos is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: cosec * cos is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cosec * cos ) `| Z) . x = ((sin . x) * (cos . (cos . x))) / ((sin . (cos . x)) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((cosec * cos ) `| Z) . x = ((sin . x) * (cos . (cos . x))) / ((sin . (cos . x)) ^2 ) )
assume A7: x in Z ; :: thesis: ((cosec * cos ) `| Z) . x = ((sin . x) * (cos . (cos . x))) / ((sin . (cos . x)) ^2 )
A8: cos is_differentiable_in x by SIN_COS:68;
A9: sin . (cos . x) <> 0 by A2, A7;
then cosec is_differentiable_in cos . x by Th2;
then diff (cosec * cos ),x = (diff cosec ,(cos . x)) * (diff cos ,x) by A8, FDIFF_2:13
.= (- ((cos . (cos . x)) / ((sin . (cos . x)) ^2 ))) * (diff cos ,x) by A9, Th2
.= (- (sin . x)) * (- ((cos . (cos . x)) / ((sin . (cos . x)) ^2 ))) by SIN_COS:68 ;
hence ((cosec * cos ) `| Z) . x = ((sin . x) * (cos . (cos . x))) / ((sin . (cos . x)) ^2 ) by A6, A7, FDIFF_1:def 8; :: thesis: verum

end;

hence ( cosec * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * cos ) `| Z) . x = ((sin . x) * (cos . (cos . x))) / ((sin . (cos . x)) ^2 ) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum