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: Z c= (dom (id Z)) /\ (dom ((2 * a) (#) f)) by A1, VALUED_1:def 1;
then A5: Z c= dom ((2 * 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, Th2;
then A8: (2 * a) (#) f is_differentiable_on Z by A5, FDIFF_1:20;
A9: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
A10: Z c= dom (id Z) by A4, XBOOLE_1:18;
then A11: id Z is_differentiable_on Z by A9, FDIFF_1:23;
A12: 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 A13: 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 A5, A7, FDIFF_1:20
.= (2 * a) * ((f `| Z) . x) by A7, A13, FDIFF_1:def 7
.= (2 * a) * (1 / (x - a)) by A2, A3, A6, A13, Th2
.= (2 * a) / (x - a) by XCMPLX_1:99 ;
:: 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 A14: x in Z ; :: thesis: (((id Z) + ((2 * a) (#) f)) `| Z) . x = (x + a) / (x - a)
then A15: ( 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, A11, A8, A14, FDIFF_1:18
.= (((id Z) `| Z) . x) + (diff (((2 * a) (#) f),x)) by A11, A14, FDIFF_1:def 7
.= (((id Z) `| Z) . x) + ((((2 * a) (#) f) `| Z) . x) by A8, A14, FDIFF_1:def 7
.= 1 + ((((2 * a) (#) f) `| Z) . x) by A10, A9, A14, FDIFF_1:23
.= 1 + ((2 * a) / (x - a)) by A12, A14
.= ((1 * (x - a)) + (2 * a)) / (x - a) by A15, XCMPLX_1:113
.= (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, A11, A8, FDIFF_1:18; :: thesis: verum