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

proof

let x be Real; :: thesis: ( x in Z implies ( sec is_differentiable_in x & diff (sec,x) = (sin . x) / ((cos . x) ^2) ) )
assume x in Z ; :: thesis: ( sec is_differentiable_in x & diff (sec,x) = (sin . x) / ((cos . x) ^2) )
then cos . x <> 0 by A3, RFUNCT_1:3;
hence ( sec is_differentiable_in x & diff (sec,x) = (sin . x) / ((cos . x) ^2) ) by Th1; :: thesis: verum

end;

then for x being Real st x in Z holds
sec is_differentiable_in x ;
then A5: sec is_differentiable_on Z by A3, FDIFF_1:9;
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_1:9;
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 (#) sec) `| Z) . x = ((1 / (cos . x)) / x) + (((ln . x) * (sin . x)) / ((cos . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((ln (#) sec) `| Z) . x = ((1 / (cos . x)) / x) + (((ln . x) * (sin . x)) / ((cos . x) ^2)) )
assume A10: x in Z ; :: thesis: ((ln (#) sec) `| Z) . x = ((1 / (cos . x)) / x) + (((ln . x) * (sin . x)) / ((cos . x) ^2))
then ((ln (#) sec) `| Z) . x = ((sec . x) * (diff (ln,x))) + ((ln . x) * (diff (sec,x))) by A1, A8, A5, FDIFF_1:21
.= ((sec . x) * (1 / x)) + ((ln . x) * (diff (sec,x))) by A9, A10
.= ((sec . x) * (1 / x)) + ((ln . x) * ((sin . x) / ((cos . x) ^2))) by A4, A10
.= ((1 / (cos . x)) / x) + (((ln . x) * (sin . x)) / ((cos . x) ^2)) by A3, A10, RFUNCT_1:def 2 ;
hence ((ln (#) sec) `| Z) . x = ((1 / (cos . x)) / x) + (((ln . x) * (sin . x)) / ((cos . x) ^2)) ; :: thesis: verum

end;

hence ( ln (#) sec is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) sec) `| Z) . x = ((1 / (cos . x)) / x) + (((ln . x) * (sin . x)) / ((cos . x) ^2)) ) ) by A1, A8, A5, FDIFF_1:21; :: thesis: verum