let a, b be Real; :: thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (a - b)) (#) f) & f = ln * (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x = x - b & f2 . x > 0 & a - b <> 0 ) ) holds
( (1 / (a - b)) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b)) ) )
let Z be open Subset of REAL; :: thesis: for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (a - b)) (#) f) & f = ln * (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x = x - b & f2 . x > 0 & a - b <> 0 ) ) holds
( (1 / (a - b)) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b)) ) )
let f, f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / (a - b)) (#) f) & f = ln * (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x = x - b & f2 . x > 0 & a - b <> 0 ) ) implies ( (1 / (a - b)) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b)) ) ) )
assume that
A1: Z c= dom ((1 / (a - b)) (#) f) and
A2: f = ln * (f1 / f2) and
A3: for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x = x - b & f2 . x > 0 & a - b <> 0 ) ; :: thesis: ( (1 / (a - b)) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b)) ) )
A4: ( ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x = x - b & f2 . x > 0 ) ) & Z c= dom f ) by A1, A3, VALUED_1:def 5;
then A5: f is_differentiable_on Z by A2, Th24;
for x being Real st x in Z holds
(((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b)) )
assume A6: x in Z ; :: thesis: (((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b))
then A7: a - b <> 0 by A3;
(((1 / (a - b)) (#) f) `| Z) . x = (1 / (a - b)) * (diff (f,x)) by A1, A5, A6, FDIFF_1:20
.= (1 / (a - b)) * ((f `| Z) . x) by A5, A6, FDIFF_1:def 7
.= (1 / (a - b)) * ((a - b) / ((x - a) * (x - b))) by A2, A4, A6, Th24
.= ((1 / (a - b)) * (a - b)) / ((x - a) * (x - b)) by XCMPLX_1:74
.= 1 / ((x - a) * (x - b)) by A7, XCMPLX_1:106 ;
hence (((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b)) ; :: thesis: verum

end;

hence ( (1 / (a - b)) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b)) ) ) by A1, A5, FDIFF_1:20; :: thesis: verum