let r be Real; for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
let Z be open Subset of REAL; for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
let f, f1, f2 be PartFunc of REAL,REAL; ( Z c= dom (f1 + f2) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) ) )
assume that
A1:
Z c= dom (f1 + f2)
and
A2:
for x being Real st x in Z holds
f1 . x = 1
and
A3:
f2 = (#Z 2) * f
and
A4:
for x being Real st x in Z holds
f . x = x / r
; ( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
A5:
for x being Real st x in Z holds
f1 . x = (0 * x) + 1
by A2;
A6:
Z c= (dom f1) /\ (dom f2)
by A1, VALUED_1:def 1;
then A7:
Z c= dom f1
by XBOOLE_1:18;
then A8:
f1 is_differentiable_on Z
by A5, FDIFF_1:23;
A9:
for x being Real st x in Z holds
f . x = ((1 / r) * x) + 0
proof
let x be
Real;
( x in Z implies f . x = ((1 / r) * x) + 0 )
assume
x in Z
;
f . x = ((1 / r) * x) + 0
hence f . x =
x / r
by A4
.=
((1 / r) * x) + 0
;
verum
end;
A10:
for x being Real st x in Z holds
f2 is_differentiable_in x
Z c= dom f2
by A6, XBOOLE_1:18;
then A12:
f2 is_differentiable_on Z
by A10, FDIFF_1:9;
A13:
for x being Real st x in Z holds
(f2 `| Z) . x = (2 * x) / (r ^2)
proof
let x be
Real;
( x in Z implies (f2 `| Z) . x = (2 * x) / (r ^2) )
assume A14:
x in Z
;
(f2 `| Z) . x = (2 * x) / (r ^2)
Z c= dom ((#Z 2) * f)
by A3, A6, XBOOLE_1:18;
then
for
y being
object st
y in Z holds
y in dom f
by FUNCT_1:11;
then A15:
Z c= dom f
;
then A16:
f is_differentiable_on Z
by A9, FDIFF_1:23;
then A17:
f is_differentiable_in x
by A14, FDIFF_1:9;
(f2 `| Z) . x =
diff (
((#Z 2) * f),
x)
by A3, A12, A14, FDIFF_1:def 7
.=
(2 * ((f . x) #Z (2 - 1))) * (diff (f,x))
by A17, TAYLOR_1:3
.=
(2 * (f . x)) * (diff (f,x))
by PREPOWER:35
.=
(2 * (x / r)) * (diff (f,x))
by A4, A14
.=
(2 * (x / r)) * ((f `| Z) . x)
by A14, A16, FDIFF_1:def 7
.=
(2 * (x / r)) * (1 / r)
by A9, A14, A15, FDIFF_1:23
.=
2
* ((x / r) * (1 / r))
.=
2
* ((x * 1) / (r * r))
by XCMPLX_1:76
.=
(2 * x) / (r ^2)
;
hence
(f2 `| Z) . x = (2 * x) / (r ^2)
;
verum
end;
for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2)
proof
let x be
Real;
( x in Z implies ((f1 + f2) `| Z) . x = (2 * x) / (r ^2) )
assume A18:
x in Z
;
((f1 + f2) `| Z) . x = (2 * x) / (r ^2)
then ((f1 + f2) `| Z) . x =
(diff (f1,x)) + (diff (f2,x))
by A1, A8, A12, FDIFF_1:18
.=
((f1 `| Z) . x) + (diff (f2,x))
by A8, A18, FDIFF_1:def 7
.=
((f1 `| Z) . x) + ((f2 `| Z) . x)
by A12, A18, FDIFF_1:def 7
.=
0 + ((f2 `| Z) . x)
by A7, A5, A18, FDIFF_1:23
.=
(2 * x) / (r ^2)
by A13, A18
;
hence
((f1 + f2) `| Z) . x = (2 * x) / (r ^2)
;
verum
end;
hence
( f1 + f2 is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + f2) `| Z) . x = (2 * x) / (r ^2) ) )
by A1, A8, A12, FDIFF_1:18; verum