let Z be open Subset of REAL; :: thesis: ( Z c= dom () implies ( sin (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| 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 () ; :: thesis: ( sin (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2)) ) )

then A3: Z c= () /\ () 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 end;
then A5: tan is_differentiable_on Z by ;
Z c= dom sin by ;
then A6: sin is_differentiable_on Z by ;
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 ;
hence diff (tan,x) = 1 / ((cos . x) ^2) by FDIFF_7:46; :: thesis: verum
end;
for x being Real st x in Z holds
(() `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2)) )
assume A8: x in Z ; :: thesis: (() `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2))
then (() `| Z) . x = ((diff (sin,x)) * (tan . x)) + ((sin . x) * (diff (tan,x))) by
.= ((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
.= ((sin . x) * ((cos . x) / (cos . x))) + ((sin . x) / ((cos . x) ^2))
.= ((sin . x) * 1) + ((sin . x) / ((cos . x) ^2)) by
.= (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
(() `| Z) . x = (sin . x) + ((sin . x) / ((cos . x) ^2)) ) ) by ; :: thesis: verum