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)) ) ) )
A1: for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
assume A2: 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 A3: Z c= (dom sin) /\ (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_1:9;
Z c= dom sin by A3, XBOOLE_1:18;
then A6: sin is_differentiable_on Z by A1, FDIFF_1:9;
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
((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 A8: 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 A2, A5, A6, FDIFF_1:21
.= ((cos . x) * (tan . x)) + ((sin . x) * (diff (tan,x))) by SIN_COS:64
.= ((cos . x) * (tan . x)) + ((sin . x) * (1 / ((cos . x) ^2))) by A7, A8
.= (((sin . x) / (cos . x)) * ((cos . x) / 1)) + ((sin . x) / ((cos . x) ^2)) by A4, A8, RFUNCT_1:def 1
.= ((sin . x) * ((cos . x) / (cos . x))) + ((sin . x) / ((cos . x) ^2))
.= ((sin . x) * 1) + ((sin . x) / ((cos . x) ^2)) by A4, A8, 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 A2, A5, A6, FDIFF_1:21; :: thesis: verum