let a, b be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (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 ) ) holds
( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b)) ) )
let Z be open Subset of REAL; :: thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (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 ) ) holds
( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b)) ) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (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 ) ) implies ( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b)) ) ) )
assume that
A1: Z c= dom (ln * (f1 / f2)) and
A2: for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x = x - b & f2 . x > 0 ) ; :: thesis: ( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b)) ) )
A3: for x being Real st x in Z holds
( f1 . x = (1 * x) + (- a) & f2 . x = (1 * x) + (- b) )

proof

let x be Real; :: thesis: ( x in Z implies ( f1 . x = (1 * x) + (- a) & f2 . x = (1 * x) + (- b) ) )
A4: ( (1 * x) + (- a) = (1 * x) - a & (1 * x) + (- b) = (1 * x) - b ) ;
assume x in Z ; :: thesis: ( f1 . x = (1 * x) + (- a) & f2 . x = (1 * x) + (- b) )
hence ( f1 . x = (1 * x) + (- a) & f2 . x = (1 * x) + (- b) ) by A2, A4; :: thesis: verum

end;

then A5: for x being Real st x in Z holds
f1 . x = (1 * x) + (- a) ;
for y being object st y in Z holds
y in dom (f1 / f2) by A1, FUNCT_1:11;
then Z c= dom (f1 / f2) by TARSKI:def 3;
then A6: Z c= (dom f1) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def 1;
then A7: Z c= dom f1 by XBOOLE_1:18;
then A8: f1 is_differentiable_on Z by A5, FDIFF_1:23;
A9: for x being Real st x in Z holds
f2 . x = (1 * x) + (- b) by A3;
A10: Z c= dom f2 by A6, XBOOLE_1:1;
then A11: f2 is_differentiable_on Z by A9, FDIFF_1:23;
for x being Real st x in Z holds
f2 . x <> 0 by A2;
then A12: f1 / f2 is_differentiable_on Z by A8, A11, FDIFF_2:21;
A13: for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (a - b) / ((x - b) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((f1 / f2) `| Z) . x = (a - b) / ((x - b) ^2) )
assume A14: x in Z ; :: thesis: ((f1 / f2) `| Z) . x = (a - b) / ((x - b) ^2)
then A15: f2 . x <> 0 by A2;
A16: ( f1 . x = x - a & f2 . x = x - b ) by A2, A14;
( f1 is_differentiable_in x & f2 is_differentiable_in x ) by A8, A11, A14, FDIFF_1:9;
then diff ((f1 / f2),x) = (((diff (f1,x)) * (f2 . x)) - ((diff (f2,x)) * (f1 . x))) / ((f2 . x) ^2) by A15, FDIFF_2:14
.= ((((f1 `| Z) . x) * (f2 . x)) - ((diff (f2,x)) * (f1 . x))) / ((f2 . x) ^2) by A8, A14, FDIFF_1:def 7
.= ((((f1 `| Z) . x) * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2) by A11, A14, FDIFF_1:def 7
.= ((1 * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2) by A7, A5, A14, FDIFF_1:23
.= ((1 * (f2 . x)) - (1 * (f1 . x))) / ((f2 . x) ^2) by A10, A9, A14, FDIFF_1:23
.= (a - b) / ((x - b) ^2) by A16 ;
hence ((f1 / f2) `| Z) . x = (a - b) / ((x - b) ^2) by A12, A14, FDIFF_1:def 7; :: thesis: verum

end;

A17: for x being Real st x in Z holds
(f1 / f2) . x > 0

proof

let x be Real; :: thesis: ( x in Z implies (f1 / f2) . x > 0 )
assume A18: x in Z ; :: thesis: (f1 / f2) . x > 0
then x in dom (f1 / f2) by A1, FUNCT_1:11;
then A19: (f1 / f2) . x = (f1 . x) * ((f2 . x) ") by RFUNCT_1:def 1
.= (f1 . x) / (f2 . x) by XCMPLX_0:def 9 ;
( f1 . x > 0 & f2 . x > 0 ) by A2, A18;
hence (f1 / f2) . x > 0 by A19, XREAL_1:139; :: thesis: verum

end;

A20: for x being Real st x in Z holds
ln * (f1 / f2) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * (f1 / f2) is_differentiable_in x )
assume x in Z ; :: thesis: ln * (f1 / f2) is_differentiable_in x
then ( f1 / f2 is_differentiable_in x & (f1 / f2) . x > 0 ) by A12, A17, FDIFF_1:9;
hence ln * (f1 / f2) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;

then A21: ln * (f1 / f2) is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b))

proof

let x be Real; :: thesis: ( x in Z implies ((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b)) )
assume A22: x in Z ; :: thesis: ((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b))
then A23: ( f2 . x = x - b & f2 . x > 0 ) by A2;
A24: ( f1 . x = x - a & f2 . x = x - b ) by A2, A22;
A25: x in dom (f1 / f2) by A1, A22, FUNCT_1:11;
( f1 / f2 is_differentiable_in x & (f1 / f2) . x > 0 ) by A12, A17, A22, FDIFF_1:9;
then diff ((ln * (f1 / f2)),x) = (diff ((f1 / f2),x)) / ((f1 / f2) . x) by TAYLOR_1:20
.= (((f1 / f2) `| Z) . x) / ((f1 / f2) . x) by A12, A22, FDIFF_1:def 7
.= ((a - b) / ((x - b) ^2)) / ((f1 / f2) . x) by A13, A22
.= ((a - b) / ((x - b) ^2)) / ((f1 . x) * ((f2 . x) ")) by A25, RFUNCT_1:def 1
.= ((a - b) / ((x - b) * (x - b))) / ((x - a) / (x - b)) by A24, XCMPLX_0:def 9
.= (((a - b) / (x - b)) / (x - b)) / ((x - a) / (x - b)) by XCMPLX_1:78
.= (((a - b) / (x - b)) / ((x - a) / (x - b))) / (x - b) by XCMPLX_1:48
.= ((a - b) / (x - a)) / (x - b) by A23, XCMPLX_1:55
.= (a - b) / ((x - a) * (x - b)) by XCMPLX_1:78 ;
hence ((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b)) by A21, A22, FDIFF_1:def 7; :: thesis: verum

end;

hence ( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = (a - b) / ((x - a) * (x - b)) ) ) by A1, A20, FDIFF_1:9; :: thesis: verum