let n be Nat; for h, x being Real
for f1, f2 being Function of REAL,REAL holds ((cdif ((f1 + f2),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (((cdif (f2,h)) . (n + 1)) . x)
let h, x be Real; for f1, f2 being Function of REAL,REAL holds ((cdif ((f1 + f2),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (((cdif (f2,h)) . (n + 1)) . x)
let f1, f2 be Function of REAL,REAL; ((cdif ((f1 + f2),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (((cdif (f2,h)) . (n + 1)) . x)
defpred S1[ Nat] means for x being Real holds ((cdif ((f1 + f2),h)) . ($1 + 1)) . x = (((cdif (f1,h)) . ($1 + 1)) . x) + (((cdif (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
((cdif ((f1 + f2),h)) . (k + 1)) . x = (((cdif (f1,h)) . (k + 1)) . x) + (((cdif (f2,h)) . (k + 1)) . x)
;
S1[k + 1]
let x be
Real;
((cdif ((f1 + f2),h)) . ((k + 1) + 1)) . x = (((cdif (f1,h)) . ((k + 1) + 1)) . x) + (((cdif (f2,h)) . ((k + 1) + 1)) . x)
A3:
(
((cdif ((f1 + f2),h)) . (k + 1)) . (x - (h / 2)) = (((cdif (f1,h)) . (k + 1)) . (x - (h / 2))) + (((cdif (f2,h)) . (k + 1)) . (x - (h / 2))) &
((cdif ((f1 + f2),h)) . (k + 1)) . (x + (h / 2)) = (((cdif (f1,h)) . (k + 1)) . (x + (h / 2))) + (((cdif (f2,h)) . (k + 1)) . (x + (h / 2))) )
by A2;
A4:
(cdif ((f1 + f2),h)) . (k + 1) is
Function of
REAL,
REAL
by Th19;
A5:
(cdif (f2,h)) . (k + 1) is
Function of
REAL,
REAL
by Th19;
A6:
(cdif (f1,h)) . (k + 1) is
Function of
REAL,
REAL
by Th19;
((cdif ((f1 + f2),h)) . ((k + 1) + 1)) . x =
(cD (((cdif ((f1 + f2),h)) . (k + 1)),h)) . x
by Def8
.=
(((cdif ((f1 + f2),h)) . (k + 1)) . (x + (h / 2))) - (((cdif ((f1 + f2),h)) . (k + 1)) . (x - (h / 2)))
by A4, Th5
.=
((((cdif (f1,h)) . (k + 1)) . (x + (h / 2))) - (((cdif (f1,h)) . (k + 1)) . (x - (h / 2)))) + ((((cdif (f2,h)) . (k + 1)) . (x + (h / 2))) - (((cdif (f2,h)) . (k + 1)) . (x - (h / 2))))
by A3
.=
((cD (((cdif (f1,h)) . (k + 1)),h)) . x) + ((((cdif (f2,h)) . (k + 1)) . (x + (h / 2))) - (((cdif (f2,h)) . (k + 1)) . (x - (h / 2))))
by A6, Th5
.=
((cD (((cdif (f1,h)) . (k + 1)),h)) . x) + ((cD (((cdif (f2,h)) . (k + 1)),h)) . x)
by A5, Th5
.=
(((cdif (f1,h)) . ((k + 1) + 1)) . x) + ((cD (((cdif (f2,h)) . (k + 1)),h)) . x)
by Def8
.=
(((cdif (f1,h)) . ((k + 1) + 1)) . x) + (((cdif (f2,h)) . ((k + 1) + 1)) . x)
by Def8
;
hence
((cdif ((f1 + f2),h)) . ((k + 1) + 1)) . x = (((cdif (f1,h)) . ((k + 1) + 1)) . x) + (((cdif (f2,h)) . ((k + 1) + 1)) . x)
;
verum
end;
A7:
S1[ 0 ]
proof
let x be
Real;
((cdif ((f1 + f2),h)) . (0 + 1)) . x = (((cdif (f1,h)) . (0 + 1)) . x) + (((cdif (f2,h)) . (0 + 1)) . x)
reconsider xx =
x,
hp =
h / 2 as
Element of
REAL by XREAL_0:def 1;
((cdif ((f1 + f2),h)) . (0 + 1)) . x =
(cD (((cdif ((f1 + f2),h)) . 0),h)) . x
by Def8
.=
(cD ((f1 + f2),h)) . x
by Def8
.=
((f1 + f2) . (x + (h / 2))) - ((f1 + f2) . (x - (h / 2)))
by Th5
.=
((f1 . (xx + (h / 2))) + (f2 . (xx + hp))) - ((f1 + f2) . (xx - hp))
by VALUED_1:1
.=
((f1 . (x + (h / 2))) + (f2 . (x + hp))) - ((f1 . (x - (h / 2))) + (f2 . (x - hp)))
by VALUED_1:1
.=
((f1 . (x + (h / 2))) - (f1 . (x - (h / 2)))) + ((f2 . (x + (h / 2))) - (f2 . (x - (h / 2))))
.=
((cD (f1,h)) . x) + ((f2 . (x + (h / 2))) - (f2 . (x - (h / 2))))
by Th5
.=
((cD (f1,h)) . x) + ((cD (f2,h)) . x)
by Th5
.=
((cD (((cdif (f1,h)) . 0),h)) . x) + ((cD (f2,h)) . x)
by Def8
.=
((cD (((cdif (f1,h)) . 0),h)) . x) + ((cD (((cdif (f2,h)) . 0),h)) . x)
by Def8
.=
(((cdif (f1,h)) . (0 + 1)) . x) + ((cD (((cdif (f2,h)) . 0),h)) . x)
by Def8
.=
(((cdif (f1,h)) . (0 + 1)) . x) + (((cdif (f2,h)) . (0 + 1)) . x)
by Def8
;
hence
((cdif ((f1 + f2),h)) . (0 + 1)) . x = (((cdif (f1,h)) . (0 + 1)) . x) + (((cdif (f2,h)) . (0 + 1)) . x)
;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(A7, A1);
hence
((cdif ((f1 + f2),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) + (((cdif (f2,h)) . (n + 1)) . x)
; verum