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_1:11;
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:63;
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_1:9;
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)) )
A6: cos is_differentiable_in tan . x by SIN_COS:63;
assume A7: x in Z ; :: thesis: ((cos * tan) `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2))
then A8: cos . x <> 0 by A2;
then tan is_differentiable_in x by FDIFF_7:46;
then diff ((cos * tan),x) = (diff (cos,(tan . x))) * (diff (tan,x)) by A6, FDIFF_2:13
.= (- (sin (tan . x))) * (diff (tan,x)) by SIN_COS:63
.= (- (sin (tan . x))) * (1 / ((cos . x) ^2)) by A8, FDIFF_7:46
.= - ((sin (tan . x)) / ((cos . x) ^2)) ;
hence ((cos * tan) `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2)) by A5, A7, FDIFF_1:def 7; :: 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_1:9; :: thesis: verum