let a be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f1 + f2) / (f1 - f2)) & ( for x being Real st x in Z holds
f1 . x = a ) & f2 = #Z 2 & ( for x being Real st x in Z holds
(f1 - f2) . x > 0 ) holds
( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) ) )
let Z be open Subset of REAL; :: thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f1 + f2) / (f1 - f2)) & ( for x being Real st x in Z holds
f1 . x = a ) & f2 = #Z 2 & ( for x being Real st x in Z holds
(f1 - f2) . x > 0 ) holds
( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) ) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((f1 + f2) / (f1 - f2)) & ( for x being Real st x in Z holds
f1 . x = a ) & f2 = #Z 2 & ( for x being Real st x in Z holds
(f1 - f2) . x > 0 ) implies ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) ) ) )
assume that
A1: Z c= dom ((f1 + f2) / (f1 - f2)) and
A2: for x being Real st x in Z holds
f1 . x = a and
A3: f2 = #Z 2 and
A4: for x being Real st x in Z holds
(f1 - f2) . x > 0 ; :: thesis: ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) ) )
A5: Z c= (dom (f1 + f2)) /\ ((dom (f1 - f2)) \ ((f1 - f2) " {0})) by A1, RFUNCT_1:def 1;
then A6: Z c= dom (f1 + f2) by XBOOLE_1:18;
then A7: f1 + f2 is_differentiable_on Z by A2, A3, Lm1;
A8: Z c= dom (f1 - f2) by A5, XBOOLE_1:1;
then A9: f1 - f2 is_differentiable_on Z by A2, A3, Lm2;
A10: for x being Real st x in Z holds
(f1 - f2) . x <> 0 by A4;
then A11: (f1 + f2) / (f1 - f2) is_differentiable_on Z by A7, A9, FDIFF_2:21;
for x being Real st x in Z holds
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2)

proof

let x be Real; :: thesis: ( x in Z implies (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) )
assume A12: x in Z ; :: thesis: (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2)
then A13: f1 . x = a by A2;
A14: (f1 - f2) . x <> 0 by A4, A12;
( f1 + f2 is_differentiable_in x & f1 - f2 is_differentiable_in x ) by A7, A9, A12, FDIFF_1:9;
then diff (((f1 + f2) / (f1 - f2)),x) = (((diff ((f1 + f2),x)) * ((f1 - f2) . x)) - ((diff ((f1 - f2),x)) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2) by A14, FDIFF_2:14
.= (((diff ((f1 + f2),x)) * ((f1 . x) - (f2 . x))) - ((diff ((f1 - f2),x)) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2) by A8, A12, VALUED_1:13
.= (((diff ((f1 + f2),x)) * ((f1 . x) - (f2 . x))) - ((diff ((f1 - f2),x)) * ((f1 . x) + (f2 . x)))) / (((f1 - f2) . x) ^2) by A6, A12, VALUED_1:def 1
.= (((diff ((f1 + f2),x)) * ((f1 . x) - (f2 . x))) - ((diff ((f1 - f2),x)) * ((f1 . x) + (f2 . x)))) / (((f1 . x) - (f2 . x)) ^2) by A8, A12, VALUED_1:13
.= (((((f1 + f2) `| Z) . x) * ((f1 . x) - (f2 . x))) - ((diff ((f1 - f2),x)) * ((f1 . x) + (f2 . x)))) / (((f1 . x) - (f2 . x)) ^2) by A7, A12, FDIFF_1:def 7
.= (((2 * x) * ((f1 . x) - (f2 . x))) - ((diff ((f1 - f2),x)) * ((f1 . x) + (f2 . x)))) / (((f1 . x) - (f2 . x)) ^2) by A2, A3, A6, A12, Lm1
.= (((2 * x) * ((f1 . x) - (f2 . x))) - ((((f1 - f2) `| Z) . x) * ((f1 . x) + (f2 . x)))) / (((f1 . x) - (f2 . x)) ^2) by A9, A12, FDIFF_1:def 7
.= (((2 * x) * ((f1 . x) - (f2 . x))) - ((- (2 * x)) * ((f1 . x) + (f2 . x)))) / (((f1 . x) - (f2 . x)) ^2) by A2, A3, A8, A12, Lm2
.= ((4 * x) * a) / ((a - (x #Z 2)) ^2) by A3, A13, TAYLOR_1:def 1
.= ((4 * x) * a) / ((a - (x |^ 2)) ^2) by PREPOWER:36
.= ((4 * x) * a) / (((a - (x |^ 2)) |^ 1) * (a - (x |^ 2)))
.= ((4 * x) * a) / ((a - (x |^ 2)) |^ (1 + 1)) by NEWTON:6
.= ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) ;
hence (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) by A11, A12, FDIFF_1:def 7; :: thesis: verum

end;

hence ( (f1 + f2) / (f1 - f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) ) ) by A7, A9, A10, FDIFF_2:21; :: thesis: verum