let Z be open Subset of REAL; :: thesis: ( Z c= dom () implies ( ln (#) exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (() / x) + ((ln . x) * ()) ) ) )

A1: for x being Real st x in Z holds
exp_R is_differentiable_in x by SIN_COS:65;
assume A2: Z c= dom () ; :: thesis: ( ln (#) exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (() / x) + ((ln . x) * ()) ) )

then A3: Z c= () /\ () by VALUED_1:def 4;
then Z c= dom exp_R by XBOOLE_1:18;
then A4: exp_R is_differentiable_on Z by ;
A5: Z c= dom ln by ;
A6: 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 ;
then x in { g where g is Real : 0 < g } by XXREAL_1:230;
then ex g being Real st
( x = g & 0 < g ) ;
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 A7: ln is_differentiable_on Z by ;
A8: 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 A6;
then x in { g where g is Real : 0 < g } ;
then x in right_open_halfline 0 by XXREAL_1:230;
hence diff (ln,x) = 1 / x by TAYLOR_1:18; :: thesis: verum
end;
for x being Real st x in Z holds
(() `| Z) . x = (() / x) + ((ln . x) * ())
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (() / x) + ((ln . x) * ()) )
assume A9: x in Z ; :: thesis: (() `| Z) . x = (() / x) + ((ln . x) * ())
then (() `| Z) . x = (() * (diff (ln,x))) + ((ln . x) * (diff (exp_R,x))) by
.= (() * (1 / x)) + ((ln . x) * (diff (exp_R,x))) by A8, A9
.= (() * (1 / x)) + ((ln . x) * ()) by SIN_COS:65 ;
hence ((ln (#) exp_R) `| Z) . x = (() / x) + ((ln . x) * ()) ; :: thesis: verum
end;
hence ( ln (#) exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (() / x) + ((ln . x) * ()) ) ) by ; :: thesis: verum