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) & ( for x being Real st x in Z holds
( f1 . x = a + x & f2 . x = a - x & f2 . x <> 0 ) ) holds
( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2) ) )
let Z be open Subset of REAL; for f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = a + x & f2 . x = a - x & f2 . x <> 0 ) ) holds
( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2) ) )
let f1, f2 be PartFunc of REAL,REAL; ( Z c= dom (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = a + x & f2 . x = a - x & f2 . x <> 0 ) ) implies ( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2) ) ) )
assume that
A1:
Z c= dom (f1 / f2)
and
A2:
for x being Real st x in Z holds
( f1 . x = a + x & f2 . x = a - x & f2 . x <> 0 )
; ( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2) ) )
A3:
Z c= (dom f1) /\ ((dom f2) \ (f2 " {0}))
by A1, RFUNCT_1:def 1;
then A4:
Z c= dom f1
by XBOOLE_1:18;
A5:
for x being Real st x in Z holds
f1 . x = (1 * x) + a
by A2;
then A6:
f1 is_differentiable_on Z
by A4, FDIFF_1:23;
A7:
for x being Real st x in Z holds
f2 . x = ((- 1) * x) + a
proof
let x be
Real;
( x in Z implies f2 . x = ((- 1) * x) + a )
assume
x in Z
;
f2 . x = ((- 1) * x) + a
then
f2 . x = a - x
by A2;
hence
f2 . x = ((- 1) * x) + a
;
verum
end;
A8:
Z c= dom f2
by A3, XBOOLE_1:1;
then A9:
f2 is_differentiable_on Z
by A7, FDIFF_1:23;
A10:
for x being Real st x in Z holds
f2 . x <> 0
by A2;
then A11:
f1 / f2 is_differentiable_on Z
by A6, A9, FDIFF_2:21;
for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2)
proof
let x be
Real;
( x in Z implies ((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2) )
assume A12:
x in Z
;
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2)
then A13:
f2 . x <> 0
by A2;
A14:
(
f1 . x = a + x &
f2 . x = a - x )
by A2, A12;
(
f1 is_differentiable_in x &
f2 is_differentiable_in x )
by A6, A9, A12, FDIFF_1:9;
then diff (
(f1 / f2),
x) =
(((diff (f1,x)) * (f2 . x)) - ((diff (f2,x)) * (f1 . x))) / ((f2 . x) ^2)
by A13, FDIFF_2:14
.=
((((f1 `| Z) . x) * (f2 . x)) - ((diff (f2,x)) * (f1 . x))) / ((f2 . x) ^2)
by A6, A12, FDIFF_1:def 7
.=
((((f1 `| Z) . x) * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2)
by A9, A12, FDIFF_1:def 7
.=
((1 * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2)
by A4, A5, A12, FDIFF_1:23
.=
((1 * (f2 . x)) - ((- 1) * (f1 . x))) / ((f2 . x) ^2)
by A8, A7, A12, FDIFF_1:23
.=
(2 * a) / ((a - x) ^2)
by A14
;
hence
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2)
by A11, A12, FDIFF_1:def 7;
verum
end;
hence
( f1 / f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 / f2) `| Z) . x = (2 * a) / ((a - x) ^2) ) )
by A6, A9, A10, FDIFF_2:21; verum