let Z be open Subset of REAL ; :: thesis: ( Z c= dom (ln (#) tan ) implies ( ln (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) tan ) `| Z) . x = (((sin . x) / (cos . x)) / x) + ((ln . x) / ((cos . x) ^2 )) ) ) )
assume A1: Z c= dom (ln (#) tan ) ; :: thesis: ( ln (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) tan ) `| Z) . x = (((sin . x) / (cos . x)) / x) + ((ln . x) / ((cos . x) ^2 )) ) )
then A2: Z c= (dom ln ) /\ (dom tan ) by VALUED_1:def 4;
then A3: Z c= dom tan by XBOOLE_1:18;
A4: for x being Real st x in Z holds
( tan is_differentiable_in x & diff tan ,x = 1 / ((cos . x) ^2 ) )

proof

let x be Real; :: thesis: ( x in Z implies ( tan is_differentiable_in x & diff tan ,x = 1 / ((cos . x) ^2 ) ) )
assume x in Z ; :: thesis: ( tan is_differentiable_in x & diff tan ,x = 1 / ((cos . x) ^2 ) )
then cos . x <> 0 by A3, Th1;
hence ( tan is_differentiable_in x & diff tan ,x = 1 / ((cos . x) ^2 ) ) by FDIFF_7:46; :: thesis: verum

end;

then for x being Real st x in Z holds
tan is_differentiable_in x ;
then A5: tan is_differentiable_on Z by A3, FDIFF_1:16;
A6: Z c= dom ln by A2, XBOOLE_1:18;
A7: for x being Real st x in Z holds
x > 0

proof

let x be Real; :: thesis: ( x in Z implies x > 0 )
assume x in Z ; :: thesis: x > 0
then x in right_open_halfline 0 by A6, TAYLOR_1:18;
then ex g being Real st
( x = g & 0 < g ) by Lm1;
hence x > 0 ; :: thesis: verum

end;

then for x being Real st x in Z holds
ln is_differentiable_in x by TAYLOR_1:18;
then A8: ln is_differentiable_on Z by A6, FDIFF_1:16;
A9: for x being Real st x in Z holds
diff ln ,x = 1 / x

proof

let x be Real; :: thesis: ( x in Z implies diff ln ,x = 1 / x )
assume x in Z ; :: thesis: diff ln ,x = 1 / x
then x > 0 by A7;
then x in right_open_halfline 0 by Lm1;
hence diff ln ,x = 1 / x by TAYLOR_1:18; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((ln (#) tan ) `| Z) . x = (((sin . x) / (cos . x)) / x) + ((ln . x) / ((cos . x) ^2 )) )
assume A10: x in Z ; :: thesis: ((ln (#) tan ) `| Z) . x = (((sin . x) / (cos . x)) / x) + ((ln . x) / ((cos . x) ^2 ))
then ((ln (#) tan ) `| Z) . x = ((tan . x) * (diff ln ,x)) + ((ln . x) * (diff tan ,x)) by A1, A8, A5, FDIFF_1:29
.= ((tan . x) * (1 / x)) + ((ln . x) * (diff tan ,x)) by A9, A10
.= ((tan . x) * (1 / x)) + ((ln . x) * (1 / ((cos . x) ^2 ))) by A4, A10
.= (((sin . x) / (cos . x)) / x) + ((ln . x) / ((cos . x) ^2 )) by A3, A10, RFUNCT_1:def 4 ;
hence ((ln (#) tan ) `| Z) . x = (((sin . x) / (cos . x)) / x) + ((ln . x) / ((cos . x) ^2 )) ; :: thesis: verum

end;

hence ( ln (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) tan ) `| Z) . x = (((sin . x) / (cos . x)) / x) + ((ln . x) / ((cos . x) ^2 )) ) ) by A1, A8, A5, FDIFF_1:29; :: thesis: verum