let a be Real; :: thesis: for Z being open Subset of REAL
for f, f1 being PartFunc of REAL ,REAL st Z c= dom ((id Z) - (a (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) holds
( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) )
let Z be open Subset of REAL ; :: thesis: for f, f1 being PartFunc of REAL ,REAL st Z c= dom ((id Z) - (a (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) holds
( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) )
let f, f1 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom ((id Z) - (a (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) implies ( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) ) )
assume that
A1: Z c= dom ((id Z) - (a (#) f)) and
A2: f = ln * f1 and
A3: for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ; :: thesis: ( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) )
A4: Z c= (dom (id Z)) /\ (dom (a (#) f)) by A1, VALUED_1:12;
then A5: Z c= dom (a (#) f) by XBOOLE_1:18;
then A6: Z c= dom (ln * f1) by A2, VALUED_1:def 5;
then A7: f is_differentiable_on Z by A2, A3, Th1;
then A8: a (#) f is_differentiable_on Z by A5, FDIFF_1:28;
A9: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
A10: Z c= dom (id Z) by A4, XBOOLE_1:18;
then A11: id Z is_differentiable_on Z by A9, FDIFF_1:31;
A12: for x being Real st x in Z holds
((a (#) f) `| Z) . x = a / (a + x)

proof

let x be Real; :: thesis: ( x in Z implies ((a (#) f) `| Z) . x = a / (a + x) )
assume A13: x in Z ; :: thesis: ((a (#) f) `| Z) . x = a / (a + x)
hence ((a (#) f) `| Z) . x = a * (diff f,x) by A5, A7, FDIFF_1:28
.= a * ((f `| Z) . x) by A7, A13, FDIFF_1:def 8
.= a * (1 / (a + x)) by A2, A3, A6, A13, Th1
.= a / (a + x) by XCMPLX_1:100 ;
:: thesis: verum

end;

for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x)

proof

let x be Real; :: thesis: ( x in Z implies (((id Z) - (a (#) f)) `| Z) . x = x / (a + x) )
assume A14: x in Z ; :: thesis: (((id Z) - (a (#) f)) `| Z) . x = x / (a + x)
then A15: ( f1 . x = a + x & f1 . x > 0 ) by A3;
(((id Z) - (a (#) f)) `| Z) . x = (diff (id Z),x) - (diff (a (#) f),x) by A1, A11, A8, A14, FDIFF_1:27
.= (((id Z) `| Z) . x) - (diff (a (#) f),x) by A11, A14, FDIFF_1:def 8
.= (((id Z) `| Z) . x) - (((a (#) f) `| Z) . x) by A8, A14, FDIFF_1:def 8
.= 1 - (((a (#) f) `| Z) . x) by A10, A9, A14, FDIFF_1:31
.= 1 - (a / (a + x)) by A12, A14
.= ((1 * (a + x)) - a) / (a + x) by A15, XCMPLX_1:128
.= x / (a + x) ;
hence (((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ; :: thesis: verum

end;

hence ( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) ) by A1, A11, A8, FDIFF_1:27; :: thesis: verum