let a be Real; for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (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 ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) )
let Z be open Subset of REAL; for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (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 ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) )
let f1, f2 be PartFunc of REAL,REAL; ( Z c= dom ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (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 ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) ) )
assume that
A1:
Z c= dom ((f1 + f2) / (f1 - f2))
and
A2:
f2 = #Z 2
and
A3:
for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x <> 0 )
; ( (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 ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) )
A4:
for x being Real st x in Z holds
f1 . x = a ^2
by A3;
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, A4, FDIFF_4:17;
A8:
Z c= dom (f1 - f2)
by A5, XBOOLE_1:1;
then A9:
f1 - f2 is_differentiable_on Z
by A2, A4, Th3;
A10:
for x being Real st x in Z holds
(f1 - f2) . x <> 0
by A3;
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 ^2)) * x) / (((a ^2) - (x |^ 2)) ^2)
proof
let x be
Real;
( x in Z implies (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) )
A12:
f2 . x =
x #Z 2
by A2, TAYLOR_1:def 1
.=
x |^ 2
by PREPOWER:36
;
assume A13:
x in Z
;
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2)
then A14:
(f1 - f2) . x <> 0
by A3;
A15:
(f1 - f2) . x =
(f1 . x) - (f2 . x)
by A8, A13, VALUED_1:13
.=
(a ^2) - (x |^ 2)
by A3, A13, A12
;
A16:
(f1 + f2) . x =
(f1 . x) + (f2 . x)
by A6, A13, VALUED_1:def 1
.=
(a ^2) + (x |^ 2)
by A3, A13, A12
;
(
f1 + f2 is_differentiable_in x &
f1 - f2 is_differentiable_in x )
by A7, A9, A13, 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
.=
(((((f1 + f2) `| Z) . x) * ((f1 - f2) . x)) - ((diff ((f1 - f2),x)) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2)
by A7, A13, FDIFF_1:def 7
.=
(((((f1 + f2) `| Z) . x) * ((f1 - f2) . x)) - ((((f1 - f2) `| Z) . x) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2)
by A9, A13, FDIFF_1:def 7
.=
(((2 * x) * ((f1 - f2) . x)) - ((((f1 - f2) `| Z) . x) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2)
by A2, A6, A4, A13, FDIFF_4:17
.=
(((2 * x) * ((f1 - f2) . x)) - ((- (2 * x)) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2)
by A2, A8, A4, A13, Th3
.=
((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2)
by A16, A15
;
hence
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2)) * x) / (((a ^2) - (x |^ 2)) ^2)
by A11, A13, FDIFF_1:def 7;
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 ^2)) * x) / (((a ^2) - (x |^ 2)) ^2) ) )
by A7, A9, A10, FDIFF_2:21; verum