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 . x) ^2 ) / (cos . x))) + (1 / (cos . x)) ) ) )
A1: for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:68;
assume A2: 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 . x) ^2 ) / (cos . x))) + (1 / (cos . x)) ) )
then A3: Z c= (dom cos ) /\ (dom tan ) by VALUED_1:def 4;
then A4: Z c= dom tan by XBOOLE_1:18;
for x being Real st x in Z holds
tan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan is_differentiable_in x )
assume x in Z ; :: thesis: tan is_differentiable_in x
then cos . x <> 0 by A4, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A5: tan is_differentiable_on Z by A4, FDIFF_1:16;
Z c= dom cos by A3, XBOOLE_1:18;
then A6: cos is_differentiable_on Z by A1, FDIFF_1:16;
A7: for x being Real st x in Z holds
diff tan ,x = 1 / ((cos . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies diff tan ,x = 1 / ((cos . x) ^2 ) )
assume x in Z ; :: thesis: diff tan ,x = 1 / ((cos . x) ^2 )
then cos . x <> 0 by A4, FDIFF_8:1;
hence diff tan ,x = 1 / ((cos . x) ^2 ) by FDIFF_7:46; :: thesis: verum

end;

for x being Real st x in Z holds
((cos (#) tan ) `| Z) . x = (- (((sin . x) ^2 ) / (cos . x))) + (1 / (cos . x))

proof

let x be Real; :: thesis: ( x in Z implies ((cos (#) tan ) `| Z) . x = (- (((sin . x) ^2 ) / (cos . x))) + (1 / (cos . x)) )
assume A8: x in Z ; :: thesis: ((cos (#) tan ) `| Z) . x = (- (((sin . x) ^2 ) / (cos . x))) + (1 / (cos . x))
then ((cos (#) tan ) `| Z) . x = ((diff cos ,x) * (tan . x)) + ((cos . x) * (diff tan ,x)) by A2, A5, A6, FDIFF_1:29
.= ((tan . x) * (- (sin . x))) + ((cos . x) * (diff tan ,x)) by SIN_COS:68
.= ((tan . x) * (- (sin . x))) + ((cos . x) * (1 / ((cos . x) ^2 ))) by A7, A8
.= (((sin . x) / (cos . x)) * (- ((sin . x) / 1))) + ((cos . x) / ((cos . x) ^2 )) by A4, A8, RFUNCT_1:def 4
.= (- (((sin . x) ^2 ) / (cos . x))) + (((cos . x) / (cos . x)) / (cos . x)) by XCMPLX_1:79
.= (- (((sin . x) ^2 ) / (cos . x))) + (1 / (cos . x)) by A4, A8, FDIFF_8:1, XCMPLX_1:60 ;
hence ((cos (#) tan ) `| Z) . x = (- (((sin . x) ^2 ) / (cos . x))) + (1 / (cos . x)) ; :: thesis: verum

end;

hence ( cos (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) tan ) `| Z) . x = (- (((sin . x) ^2 ) / (cos . x))) + (1 / (cos . x)) ) ) by A2, A5, A6, FDIFF_1:29; :: thesis: verum