let Z be open Subset of REAL ; :: thesis: for f, f1 being PartFunc of REAL ,REAL st Z c= dom (ln * f) & f = exp_R - (exp_R * f1) & ( for x being Real st x in Z holds
( f1 . x = - x & f . x > 0 ) ) holds
( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) )
let f, f1 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (ln * f) & f = exp_R - (exp_R * f1) & ( for x being Real st x in Z holds
( f1 . x = - x & f . x > 0 ) ) implies ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) ) )
assume that
A1: ( Z c= dom (ln * f) & f = exp_R - (exp_R * f1) ) and
A2: for x being Real st x in Z holds
( f1 . x = - x & f . x > 0 ) ; :: thesis: ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) )
A3: for x being Real st x in Z holds
f1 . x = - x by A2;
for y being set st y in Z holds
y in dom f by A1, FUNCT_1:21;
then A4: Z c= dom (exp_R - (exp_R * f1)) by A1, TARSKI:def 3;
then Z c= (dom exp_R ) /\ (dom (exp_R * f1)) by VALUED_1:12;
then A5: Z c= dom (exp_R * f1) by XBOOLE_1:18;
then A6: ( exp_R * f1 is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f1) `| Z) . x = - (exp_R (- x)) ) ) by A3, Th14;
A7: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
A8: for x being Real st x in Z holds
((exp_R - (exp_R * f1)) `| Z) . x = (exp_R x) + (exp_R (- x))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R - (exp_R * f1)) `| Z) . x = (exp_R x) + (exp_R (- x)) )
assume A9: x in Z ; :: thesis: ((exp_R - (exp_R * f1)) `| Z) . x = (exp_R x) + (exp_R (- x))
hence ((exp_R - (exp_R * f1)) `| Z) . x = (diff exp_R ,x) - (diff (exp_R * f1),x) by A4, A6, A7, FDIFF_1:27
.= (exp_R . x) - (diff (exp_R * f1),x) by SIN_COS:70
.= (exp_R . x) - (((exp_R * f1) `| Z) . x) by A6, A9, FDIFF_1:def 8
.= (exp_R . x) - (- (exp_R (- x))) by A3, A5, A9, Th14
.= (exp_R . x) + (exp_R (- x))
.= (exp_R x) + (exp_R (- x)) by SIN_COS:def 27 ;
:: thesis: verum

end;

then A10: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = (exp_R x) + (exp_R (- x)) ) ) by A1, A4, A6, A7, FDIFF_1:27;
A11: for x being Real st x in Z holds
ln * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * f is_differentiable_in x )
assume A12: x in Z ; :: thesis: ln * f is_differentiable_in x
then A13: f is_differentiable_in x by A10, FDIFF_1:16;
f . x > 0 by A2, A12;
hence ln * f is_differentiable_in x by A13, TAYLOR_1:20; :: thesis: verum

end;

then A14: ln * f is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x)))

proof

let x be Real; :: thesis: ( x in Z implies ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) )
assume A15: x in Z ; :: thesis: ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x)))
then A16: f . x = (exp_R . x) - ((exp_R * f1) . x) by A1, A4, VALUED_1:13
.= (exp_R . x) - (exp_R . (f1 . x)) by A5, A15, FUNCT_1:22
.= (exp_R . x) - (exp_R . (- x)) by A2, A15
.= (exp_R x) - (exp_R . (- x)) by SIN_COS:def 27
.= (exp_R x) - (exp_R (- x)) by SIN_COS:def 27 ;
A17: f is_differentiable_in x by A10, A15, FDIFF_1:16;
f . x > 0 by A2, A15;
then diff (ln * f),x = (diff f,x) / (f . x) by A17, TAYLOR_1:20
.= ((f `| Z) . x) / (f . x) by A10, A15, FDIFF_1:def 8
.= ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) by A1, A8, A15, A16 ;
hence ((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) by A14, A15, FDIFF_1:def 8; :: thesis: verum

end;

hence ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = ((exp_R x) + (exp_R (- x))) / ((exp_R x) - (exp_R (- x))) ) ) by A1, A11, FDIFF_1:16; :: thesis: verum