let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sin (#) tan ) implies ( sin (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) tan ) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2 )) ) ) )
assume A1: Z c= dom (sin (#) tan ) ; :: thesis: ( sin (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) tan ) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2 )) ) )
then Z c= (dom sin ) /\ (dom tan ) by VALUED_1:def 4;
then A2: ( Z c= dom sin & 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 A2, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A3: tan is_differentiable_on Z by A2, FDIFF_1:16;
for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:69;
then A4: sin is_differentiable_on Z by A2, FDIFF_1:16;
A5: 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 A2, 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
((sin (#) tan ) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((sin (#) tan ) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2 )) )
assume A6: x in Z ; :: thesis: ((sin (#) tan ) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2 ))
then ((sin (#) tan ) `| Z) . x = ((diff sin ,x) * (tan . x)) + ((sin . x) * (diff tan ,x)) by A1, A3, A4, FDIFF_1:29
.= ((cos . x) * (tan . x)) + ((sin . x) * (diff tan ,x)) by SIN_COS:69
.= ((cos . x) * (tan . x)) + ((sin . x) * (1 / ((cos . x) ^2 ))) by A5, A6
.= (((sin . x) / (cos . x)) * ((cos . x) / 1)) + ((sin . x) / ((cos . x) ^2 )) by A2, A6, RFUNCT_1:def 4
.= ((sin . x) * ((cos . x) / (cos . x))) + ((sin . x) / ((cos . x) ^2 ))
.= ((sin . x) * 1) + ((sin . x) / ((cos . x) ^2 )) by A2, A6, FDIFF_8:1, XCMPLX_1:60
.= (sin . x) + ((sin . x) / ((cos . x) ^2 )) ;
hence ((sin (#) tan ) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2 )) ; :: thesis: verum

end;

hence ( sin (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) tan ) `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2 )) ) ) by A1, A3, A4, FDIFF_1:29; :: thesis: verum