let Z be open Subset of REAL ; :: thesis: ( Z c= dom (ln (#) exp_R ) implies ( ln (#) exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) exp_R ) `| Z) . x = ((exp_R . x) / x) + ((ln . x) * (exp_R . x)) ) ) )
A1: for x being Real st x in Z holds
exp_R is_differentiable_in x by SIN_COS:70;
assume A2: Z c= dom (ln (#) exp_R ) ; :: thesis: ( ln (#) exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) exp_R ) `| Z) . x = ((exp_R . x) / x) + ((ln . x) * (exp_R . x)) ) )
then A3: Z c= (dom ln ) /\ (dom exp_R ) by VALUED_1:def 4;
then Z c= dom exp_R by XBOOLE_1:18;
then A4: exp_R is_differentiable_on Z by A1, FDIFF_1:16;
A5: Z c= dom ln by A3, XBOOLE_1:18;
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 A5, TAYLOR_1:18;
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 A5, FDIFF_1:16;
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
((ln (#) exp_R ) `| Z) . x = ((exp_R . x) / x) + ((ln . x) * (exp_R . x))

proof

let x be Real; :: thesis: ( x in Z implies ((ln (#) exp_R ) `| Z) . x = ((exp_R . x) / x) + ((ln . x) * (exp_R . x)) )
assume A9: x in Z ; :: thesis: ((ln (#) exp_R ) `| Z) . x = ((exp_R . x) / x) + ((ln . x) * (exp_R . x))
then ((ln (#) exp_R ) `| Z) . x = ((exp_R . x) * (diff ln ,x)) + ((ln . x) * (diff exp_R ,x)) by A2, A7, A4, FDIFF_1:29
.= ((exp_R . x) * (1 / x)) + ((ln . x) * (diff exp_R ,x)) by A8, A9
.= ((exp_R . x) * (1 / x)) + ((ln . x) * (exp_R . x)) by SIN_COS:70 ;
hence ((ln (#) exp_R ) `| Z) . x = ((exp_R . x) / x) + ((ln . x) * (exp_R . x)) ; :: thesis: verum

end;

hence ( ln (#) exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) exp_R ) `| Z) . x = ((exp_R . x) / x) + ((ln . x) * (exp_R . x)) ) ) by A2, A7, A4, FDIFF_1:29; :: thesis: verum