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

proof

let x be Real; :: thesis: ( x in Z implies cos . x <> 0 )
assume x in Z ; :: thesis: cos . x <> 0
then x in dom (sin / cos ) by A1, FUNCT_1:21;
hence cos . x <> 0 by FDIFF_8:1; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies cos * tan is_differentiable_in x )
assume x in Z ; :: thesis: cos * tan is_differentiable_in x
then cos . x <> 0 by A2;
then A4: tan is_differentiable_in x by FDIFF_7:46;
cos is_differentiable_in tan . x by SIN_COS:68;
hence cos * tan is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((cos * tan ) `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2 )) )
assume A6: x in Z ; :: thesis: ((cos * tan ) `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2 ))
then A7: cos . x <> 0 by A2;
then A8: tan is_differentiable_in x by FDIFF_7:46;
cos is_differentiable_in tan . x by SIN_COS:68;
then diff (cos * tan ),x = (diff cos ,(tan . x)) * (diff tan ,x) by A8, FDIFF_2:13
.= (- (sin (tan . x))) * (diff tan ,x) by SIN_COS:68
.= (- (sin (tan . x))) * (1 / ((cos . x) ^2 )) by A7, FDIFF_7:46
.= - ((sin (tan . x)) / ((cos . x) ^2 )) ;
hence ((cos * tan ) `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2 )) by A5, A6, FDIFF_1:def 8; :: thesis: verum

end;

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