let a be Real; :: thesis: for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom (((2 * a) (#) f) - (id Z)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) holds
( ((2 * a) (#) f) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x) ) )
let Z be open Subset of REAL; :: thesis: for f, f1 being PartFunc of REAL,REAL st Z c= dom (((2 * a) (#) f) - (id Z)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) holds
( ((2 * a) (#) f) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x) ) )
let f, f1 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (((2 * a) (#) f) - (id Z)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) implies ( ((2 * a) (#) f) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x) ) ) )
assume that
A1: Z c= dom (((2 * a) (#) f) - (id Z)) and
A2: f = ln * f1 and
A3: for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ; :: thesis: ( ((2 * a) (#) f) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x) ) )
A4: Z c= (dom ((2 * a) (#) f)) /\ (dom (id Z)) by A1, VALUED_1:12;
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, Th1;
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) / (a + x)

proof

let x be Real; :: thesis: ( x in Z implies (((2 * a) (#) f) `| Z) . x = (2 * a) / (a + x) )
assume A13: x in Z ; :: thesis: (((2 * a) (#) f) `| Z) . x = (2 * a) / (a + x)
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 / (a + x)) by A2, A3, A6, A13, Th1
.= (2 * a) / (a + x) by XCMPLX_1:99 ;
:: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x) )
assume A14: x in Z ; :: thesis: ((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x)
then A15: ( f1 . x = a + x & f1 . x > 0 ) by A3;
((((2 * a) (#) f) - (id Z)) `| Z) . x = (diff (((2 * a) (#) f),x)) - (diff ((id Z),x)) by A1, A11, A8, A14, FDIFF_1:19
.= (diff (((2 * a) (#) f),x)) - (((id Z) `| Z) . x) by A11, A14, FDIFF_1:def 7
.= ((((2 * a) (#) f) `| Z) . x) - (((id Z) `| Z) . x) by A8, A14, FDIFF_1:def 7
.= ((((2 * a) (#) f) `| Z) . x) - 1 by A10, A9, A14, FDIFF_1:23
.= ((2 * a) / (a + x)) - 1 by A12, A14
.= ((2 * a) - (1 * (a + x))) / (a + x) by A15, XCMPLX_1:126
.= (a - x) / (a + x) ;
hence ((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x) ; :: thesis: verum

end;

hence ( ((2 * a) (#) f) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x) ) ) by A1, A11, A8, FDIFF_1:19; :: thesis: verum