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 A1: ( 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 ) ) ; :: 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) ) )
then A2: Z c= (dom (f1 + f2)) /\ ((dom (f1 - f2)) \ ((f1 - f2) " {0 })) by RFUNCT_1:def 4;
then A3: ( Z c= dom (f1 + f2) & Z c= (dom (f1 - f2)) \ ((f1 - f2) " {0 }) ) by XBOOLE_1:18;
A4: Z c= dom (f1 - f2) by A2, XBOOLE_1:1;
A5: ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) ) by A1, A3, Lm1;
A6: ( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x) ) ) by A1, A4, Lm2;
A7: for x being Real st x in Z holds
(f1 - f2) . x <> 0 by A1;
then A8: (f1 + f2) / (f1 - f2) is_differentiable_on Z by A5, A6, 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 A9: x in Z ; :: thesis: (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2)
then A10: f1 + f2 is_differentiable_in x by A5, FDIFF_1:16;
A11: f1 - f2 is_differentiable_in x by A6, A9, FDIFF_1:16;
A12: ( f1 . x = a & f2 = #Z 2 & (f1 - f2) . x <> 0 ) by A1, A9;
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 A10, A11, FDIFF_2:14
.= (((diff (f1 + f2),x) * ((f1 . x) - (f2 . x))) - ((diff (f1 - f2),x) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2 ) by A4, A9, VALUED_1:13
.= (((diff (f1 + f2),x) * ((f1 . x) - (f2 . x))) - ((diff (f1 - f2),x) * ((f1 . x) + (f2 . x)))) / (((f1 - f2) . x) ^2 ) by A3, A9, 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 A4, A9, 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 A5, A9, FDIFF_1:def 8
.= (((2 * x) * ((f1 . x) - (f2 . x))) - ((diff (f1 - f2),x) * ((f1 . x) + (f2 . x)))) / (((f1 . x) - (f2 . x)) ^2 ) by A1, A3, A9, Lm1
.= (((2 * x) * ((f1 . x) - (f2 . x))) - ((((f1 - f2) `| Z) . x) * ((f1 . x) + (f2 . x)))) / (((f1 . x) - (f2 . x)) ^2 ) by A6, A9, FDIFF_1:def 8
.= (((2 * x) * ((f1 . x) - (f2 . x))) - ((- (2 * x)) * ((f1 . x) + (f2 . x)))) / (((f1 . x) - (f2 . x)) ^2 ) by A1, A4, A9, Lm2
.= ((4 * x) * a) / ((a - (x #Z 2)) ^2 ) by A12, TAYLOR_1:def 1
.= ((4 * x) * a) / ((a - (x |^ 2)) ^2 ) by PREPOWER:46
.= ((4 * x) * a) / (((a - (x |^ 2)) |^ 1) * (a - (x |^ 2))) by NEWTON:10
.= ((4 * x) * a) / ((a - (x |^ 2)) |^ (1 + 1)) by NEWTON:11
.= ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) ;
hence (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * a) * x) / ((a - (x |^ 2)) |^ 2) by A8, A9, FDIFF_1:def 8; :: 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 A5, A6, A7, FDIFF_2:21; :: thesis: verum