let n be Nat; for h, x being Real
for f1, f2 being Function of REAL,REAL holds ((fdif ((f1 + f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (((fdif (f2,h)) . (n + 1)) . x)
let h, x be Real; for f1, f2 being Function of REAL,REAL holds ((fdif ((f1 + f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (((fdif (f2,h)) . (n + 1)) . x)
let f1, f2 be Function of REAL,REAL; ((fdif ((f1 + f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (((fdif (f2,h)) . (n + 1)) . x)
defpred S1[ Nat] means for x being Real holds ((fdif ((f1 + f2),h)) . ($1 + 1)) . x = (((fdif (f1,h)) . ($1 + 1)) . x) + (((fdif (f2,h)) . ($1 + 1)) . x);
A1:
for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be
Nat;
( S1[k] implies S1[k + 1] )
assume A2:
for
x being
Real holds
((fdif ((f1 + f2),h)) . (k + 1)) . x = (((fdif (f1,h)) . (k + 1)) . x) + (((fdif (f2,h)) . (k + 1)) . x)
;
S1[k + 1]
let x be
Real;
((fdif ((f1 + f2),h)) . ((k + 1) + 1)) . x = (((fdif (f1,h)) . ((k + 1) + 1)) . x) + (((fdif (f2,h)) . ((k + 1) + 1)) . x)
A3:
(
((fdif ((f1 + f2),h)) . (k + 1)) . x = (((fdif (f1,h)) . (k + 1)) . x) + (((fdif (f2,h)) . (k + 1)) . x) &
((fdif ((f1 + f2),h)) . (k + 1)) . (x + h) = (((fdif (f1,h)) . (k + 1)) . (x + h)) + (((fdif (f2,h)) . (k + 1)) . (x + h)) )
by A2;
A4:
(fdif ((f1 + f2),h)) . (k + 1) is
Function of
REAL,
REAL
by Th2;
A5:
(fdif (f2,h)) . (k + 1) is
Function of
REAL,
REAL
by Th2;
A6:
(fdif (f1,h)) . (k + 1) is
Function of
REAL,
REAL
by Th2;
((fdif ((f1 + f2),h)) . ((k + 1) + 1)) . x =
(fD (((fdif ((f1 + f2),h)) . (k + 1)),h)) . x
by Def6
.=
(((fdif ((f1 + f2),h)) . (k + 1)) . (x + h)) - (((fdif ((f1 + f2),h)) . (k + 1)) . x)
by A4, Th3
.=
((((fdif (f1,h)) . (k + 1)) . (x + h)) - (((fdif (f1,h)) . (k + 1)) . x)) + ((((fdif (f2,h)) . (k + 1)) . (x + h)) - (((fdif (f2,h)) . (k + 1)) . x))
by A3
.=
((fD (((fdif (f1,h)) . (k + 1)),h)) . x) + ((((fdif (f2,h)) . (k + 1)) . (x + h)) - (((fdif (f2,h)) . (k + 1)) . x))
by A6, Th3
.=
((fD (((fdif (f1,h)) . (k + 1)),h)) . x) + ((fD (((fdif (f2,h)) . (k + 1)),h)) . x)
by A5, Th3
.=
(((fdif (f1,h)) . ((k + 1) + 1)) . x) + ((fD (((fdif (f2,h)) . (k + 1)),h)) . x)
by Def6
.=
(((fdif (f1,h)) . ((k + 1) + 1)) . x) + (((fdif (f2,h)) . ((k + 1) + 1)) . x)
by Def6
;
hence
((fdif ((f1 + f2),h)) . ((k + 1) + 1)) . x = (((fdif (f1,h)) . ((k + 1) + 1)) . x) + (((fdif (f2,h)) . ((k + 1) + 1)) . x)
;
verum
end;
A7:
S1[ 0 ]
proof
let x be
Real;
((fdif ((f1 + f2),h)) . (0 + 1)) . x = (((fdif (f1,h)) . (0 + 1)) . x) + (((fdif (f2,h)) . (0 + 1)) . x)
reconsider xx =
x,
h =
h as
Element of
REAL by XREAL_0:def 1;
((fdif ((f1 + f2),h)) . (0 + 1)) . x =
(fD (((fdif ((f1 + f2),h)) . 0),h)) . x
by Def6
.=
(fD ((f1 + f2),h)) . x
by Def6
.=
((f1 + f2) . (x + h)) - ((f1 + f2) . x)
by Th3
.=
((f1 . (xx + h)) + (f2 . (xx + h))) - ((f1 + f2) . xx)
by VALUED_1:1
.=
((f1 . (x + h)) + (f2 . (x + h))) - ((f1 . x) + (f2 . x))
by VALUED_1:1
.=
((f1 . (x + h)) - (f1 . x)) + ((f2 . (x + h)) - (f2 . x))
.=
((fD (f1,h)) . x) + ((f2 . (x + h)) - (f2 . x))
by Th3
.=
((fD (f1,h)) . x) + ((fD (f2,h)) . x)
by Th3
.=
((fD (((fdif (f1,h)) . 0),h)) . x) + ((fD (f2,h)) . x)
by Def6
.=
((fD (((fdif (f1,h)) . 0),h)) . x) + ((fD (((fdif (f2,h)) . 0),h)) . x)
by Def6
.=
(((fdif (f1,h)) . (0 + 1)) . x) + ((fD (((fdif (f2,h)) . 0),h)) . x)
by Def6
.=
(((fdif (f1,h)) . (0 + 1)) . x) + (((fdif (f2,h)) . (0 + 1)) . x)
by Def6
;
hence
((fdif ((f1 + f2),h)) . (0 + 1)) . x = (((fdif (f1,h)) . (0 + 1)) . x) + (((fdif (f2,h)) . (0 + 1)) . x)
;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(A7, A1);
hence
((fdif ((f1 + f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (((fdif (f2,h)) . (n + 1)) . x)
; verum