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)) & 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 ; :: thesis: 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 ; :: thesis: ( 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 )
; :: 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 ^2 )) * x) / (((a ^2 ) - (x |^ 2)) ^2 ) ) )
A4:
Z c= (dom (f1 + f2)) /\ ((dom (f1 - f2)) \ ((f1 - f2) " {0 }))
by A1, RFUNCT_1:def 4;
then A5:
( Z c= dom (f1 + f2) & Z c= (dom (f1 - f2)) \ ((f1 - f2) " {0 }) )
by XBOOLE_1:18;
A6:
Z c= dom (f1 - f2)
by A4, XBOOLE_1:1;
A7:
for x being Real st x in Z holds
f1 . x = a ^2
by A3;
then A8:
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = 2 * x ) )
by A2, A5, FDIFF_4:17;
A9:
( f1 - f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - f2) `| Z) . x = - (2 * x) ) )
by A2, A6, A7, 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 A8, 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;
:: thesis: ( x in Z implies (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2 )) * x) / (((a ^2 ) - (x |^ 2)) ^2 ) )
assume A12:
x in Z
;
:: thesis: (((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2 )) * x) / (((a ^2 ) - (x |^ 2)) ^2 )
then A13:
f1 + f2 is_differentiable_in x
by A8, FDIFF_1:16;
A14:
f1 - f2 is_differentiable_in x
by A9, A12, FDIFF_1:16;
A15:
f2 . x =
x #Z 2
by A2, TAYLOR_1:def 1
.=
x |^ 2
by PREPOWER:46
;
A16:
(f1 + f2) . x =
(f1 . x) + (f2 . x)
by A5, A12, VALUED_1:def 1
.=
(a ^2 ) + (x |^ 2)
by A3, A12, A15
;
A17:
(f1 - f2) . x =
(f1 . x) - (f2 . x)
by A6, A12, VALUED_1:13
.=
(a ^2 ) - (x |^ 2)
by A3, A12, A15
;
(f1 - f2) . x <> 0
by A3, A12;
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 A13, A14, FDIFF_2:14
.=
(((((f1 + f2) `| Z) . x) * ((f1 - f2) . x)) - ((diff (f1 - f2),x) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2 )
by A8, A12, FDIFF_1:def 8
.=
(((((f1 + f2) `| Z) . x) * ((f1 - f2) . x)) - ((((f1 - f2) `| Z) . x) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2 )
by A9, A12, FDIFF_1:def 8
.=
(((2 * x) * ((f1 - f2) . x)) - ((((f1 - f2) `| Z) . x) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2 )
by A2, A5, A7, A12, FDIFF_4:17
.=
(((2 * x) * ((f1 - f2) . x)) - ((- (2 * x)) * ((f1 + f2) . x))) / (((f1 - f2) . x) ^2 )
by A2, A6, A7, A12, Th3
.=
((4 * (a ^2 )) * x) / (((a ^2 ) - (x |^ 2)) ^2 )
by A16, A17
;
hence
(((f1 + f2) / (f1 - f2)) `| Z) . x = ((4 * (a ^2 )) * x) / (((a ^2 ) - (x |^ 2)) ^2 )
by A11, A12, 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 ^2 )) * x) / (((a ^2 ) - (x |^ 2)) ^2 ) ) )
by A8, A9, A10, FDIFF_2:21; :: thesis: verum