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) - ((2 * a) (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = x + a & f1 . x > 0 ) ) holds
( (id Z) - ((2 * a) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a) ) )
let Z be open Subset of REAL ; :: thesis: for f, f1 being PartFunc of REAL ,REAL st Z c= dom ((id Z) - ((2 * a) (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = x + a & f1 . x > 0 ) ) holds
( (id Z) - ((2 * a) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a) ) )
let f, f1 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom ((id Z) - ((2 * a) (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = x + a & f1 . x > 0 ) ) implies ( (id Z) - ((2 * a) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a) ) ) )
assume that
A1: Z c= dom ((id Z) - ((2 * a) (#) f)) and
A2: f = ln * f1 and
A3: for x being Real st x in Z holds
( f1 . x = x + a & f1 . x > 0 ) ; :: thesis: ( (id Z) - ((2 * a) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a) ) )
A4: for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) by A3;
A5: Z c= (dom (id Z)) /\ (dom ((2 * a) (#) f)) by A1, VALUED_1:12;
then A6: Z c= dom ((2 * a) (#) f) by XBOOLE_1:18;
then A7: Z c= dom (ln * f1) by A2, VALUED_1:def 5;
then A8: f is_differentiable_on Z by A2, A4, Th1;
then A9: (2 * a) (#) f is_differentiable_on Z by A6, FDIFF_1:28;
A10: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
A11: Z c= dom (id Z) by A5, XBOOLE_1:18;
then A12: id Z is_differentiable_on Z by A10, FDIFF_1:31;
A13: for x being Real st x in Z holds
(((2 * a) (#) f) `| Z) . x = (2 * a) / (x + a)

proof

let x be Real; :: thesis: ( x in Z implies (((2 * a) (#) f) `| Z) . x = (2 * a) / (x + a) )
assume A14: x in Z ; :: thesis: (((2 * a) (#) f) `| Z) . x = (2 * a) / (x + a)
hence (((2 * a) (#) f) `| Z) . x = (2 * a) * (diff f,x) by A6, A8, FDIFF_1:28
.= (2 * a) * ((f `| Z) . x) by A8, A14, FDIFF_1:def 8
.= (2 * a) * (1 / (x + a)) by A2, A7, A4, A14, Th1
.= (2 * a) / (x + a) by XCMPLX_1:100 ;
:: thesis: verum

end;

for x being Real st x in Z holds
(((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a)

proof

let x be Real; :: thesis: ( x in Z implies (((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a) )
assume A15: x in Z ; :: thesis: (((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a)
then A16: ( f1 . x = x + a & f1 . x > 0 ) by A3;
(((id Z) - ((2 * a) (#) f)) `| Z) . x = (diff (id Z),x) - (diff ((2 * a) (#) f),x) by A1, A12, A9, A15, FDIFF_1:27
.= (((id Z) `| Z) . x) - (diff ((2 * a) (#) f),x) by A12, A15, FDIFF_1:def 8
.= (((id Z) `| Z) . x) - ((((2 * a) (#) f) `| Z) . x) by A9, A15, FDIFF_1:def 8
.= 1 - ((((2 * a) (#) f) `| Z) . x) by A11, A10, A15, FDIFF_1:31
.= 1 - ((2 * a) / (x + a)) by A13, A15
.= ((1 * (x + a)) - (2 * a)) / (x + a) by A16, XCMPLX_1:128
.= (x - a) / (x + a) ;
hence (((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a) ; :: thesis: verum

end;

hence ( (id Z) - ((2 * a) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((2 * a) (#) f)) `| Z) . x = (x - a) / (x + a) ) ) by A1, A12, A9, FDIFF_1:27; :: thesis: verum