let a 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 + a & 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 = (2 * a) / ((x ^2) - (a ^2)) ) )
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 + a & 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 = (2 * a) / ((x ^2) - (a ^2)) ) )
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 + a & 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 = (2 * a) / ((x ^2) - (a ^2)) ) ) )
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 + a & 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 = (2 * a) / ((x ^2) - (a ^2)) ) )
A3: for x being Real st x in Z holds
f2 . x = (1 * x) + a by A2;
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 A4: Z c= (dom f1) /\ ((dom f2) \ (f2 " {0})) by RFUNCT_1:def 1;
then A5: Z c= dom f1 by XBOOLE_1:18;
A6: Z c= dom f2 by A4, XBOOLE_1:1;
then A7: f2 is_differentiable_on Z by A3, FDIFF_1:23;
A8: for x being Real st x in Z holds
f1 . x = (1 * x) + (- a)

proof

let x be Real; :: thesis: ( x in Z implies f1 . x = (1 * x) + (- a) )
A9: (1 * x) + (- a) = (1 * x) - a ;
assume x in Z ; :: thesis: f1 . x = (1 * x) + (- a)
hence f1 . x = (1 * x) + (- a) by A2, A9; :: thesis: verum

end;

then A10: f1 is_differentiable_on Z by A5, FDIFF_1:23;
for x being Real st x in Z holds
f2 . x <> 0 by A2;
then A11: f1 / f2 is_differentiable_on Z by A10, A7, FDIFF_2:21;
A12: for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((x + a) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((f1 / f2) `| Z) . x = (2 * a) / ((x + a) ^2) )
assume A13: x in Z ; :: thesis: ((f1 / f2) `| Z) . x = (2 * a) / ((x + a) ^2)
then A14: f2 . x <> 0 by A2;
A15: ( f1 . x = x - a & f2 . x = x + a ) by A2, A13;
( f1 is_differentiable_in x & f2 is_differentiable_in x ) by A10, A7, A13, FDIFF_1:9;
then diff ((f1 / f2),x) = (((diff (f1,x)) * (f2 . x)) - ((diff (f2,x)) * (f1 . x))) / ((f2 . x) ^2) by A14, FDIFF_2:14
.= ((((f1 `| Z) . x) * (f2 . x)) - ((diff (f2,x)) * (f1 . x))) / ((f2 . x) ^2) by A10, A13, FDIFF_1:def 7
.= ((((f1 `| Z) . x) * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2) by A7, A13, FDIFF_1:def 7
.= ((1 * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2) by A5, A8, A13, FDIFF_1:23
.= ((1 * (f2 . x)) - (1 * (f1 . x))) / ((f2 . x) ^2) by A6, A3, A13, FDIFF_1:23
.= (2 * a) / ((x + a) ^2) by A15 ;
hence ((f1 / f2) `| Z) . x = (2 * a) / ((x + a) ^2) by A11, A13, FDIFF_1:def 7; :: thesis: verum

end;

A16: 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 A17: x in Z ; :: thesis: (f1 / f2) . x > 0
then x in dom (f1 / f2) by A1, FUNCT_1:11;
then A18: (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, A17;
hence (f1 / f2) . x > 0 by A18, XREAL_1:139; :: thesis: verum

end;

A19: 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 A11, A16, FDIFF_1:9;
hence ln * (f1 / f2) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;

then A20: 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 = (2 * a) / ((x ^2) - (a ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((ln * (f1 / f2)) `| Z) . x = (2 * a) / ((x ^2) - (a ^2)) )
assume A21: x in Z ; :: thesis: ((ln * (f1 / f2)) `| Z) . x = (2 * a) / ((x ^2) - (a ^2))
then A22: ( f2 . x = x + a & f2 . x > 0 ) by A2;
A23: ( f1 . x = x - a & f2 . x = x + a ) by A2, A21;
A24: x in dom (f1 / f2) by A1, A21, FUNCT_1:11;
( f1 / f2 is_differentiable_in x & (f1 / f2) . x > 0 ) by A11, A16, A21, 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 A11, A21, FDIFF_1:def 7
.= ((2 * a) / ((x + a) ^2)) / ((f1 / f2) . x) by A12, A21
.= ((2 * a) / ((x + a) ^2)) / ((f1 . x) * ((f2 . x) ")) by A24, RFUNCT_1:def 1
.= ((2 * a) / ((x + a) * (x + a))) / ((x - a) / (x + a)) by A23, XCMPLX_0:def 9
.= (((2 * a) / (x + a)) / (x + a)) / ((x - a) / (x + a)) by XCMPLX_1:78
.= (((2 * a) / (x + a)) / ((x - a) / (x + a))) / (x + a) by XCMPLX_1:48
.= ((2 * a) / (x - a)) / (x + a) by A22, XCMPLX_1:55
.= (2 * a) / ((x - a) * (x + a)) by XCMPLX_1:78
.= (2 * a) / ((x ^2) - (a ^2)) ;
hence ((ln * (f1 / f2)) `| Z) . x = (2 * a) / ((x ^2) - (a ^2)) by A20, A21, 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 = (2 * a) / ((x ^2) - (a ^2)) ) ) by A1, A19, FDIFF_1:9; :: thesis: verum