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:
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)
x - (h / 2) in REAL
by XREAL_0:def 1;
then
(
x - (h / 2) in dom f1 &
x - (h / 2) in dom f2 )
by FUNCT_2:def 1;
then
x - (h / 2) in (dom f1) /\ (dom f2)
by XBOOLE_0:def 4;
then A2:
x - (h / 2) in dom (f1 - f2)
by VALUED_1:12;
x + (h / 2) in REAL
by XREAL_0:def 1;
then
(
x + (h / 2) in dom f1 &
x + (h / 2) in dom f2 )
by FUNCT_2:def 1;
then
x + (h / 2) in (dom f1) /\ (dom f2)
by XBOOLE_0:def 4;
then A3:
x + (h / 2) in dom (f1 - f2)
by VALUED_1:12;
((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 . (x + (h / 2))) - (f2 . (x + (h / 2)))) - ((f1 - f2) . (x - (h / 2)))
by A3, VALUED_1:13
.=
((f1 . (x + (h / 2))) - (f2 . (x + (h / 2)))) - ((f1 . (x - (h / 2))) - (f2 . (x - (h / 2))))
by A2, VALUED_1:13
.=
((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;
A4:
for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be
Nat;
( S1[k] implies S1[k + 1] )
assume A5:
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)
A6:
(
((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 A5;
A7:
(cdif ((f1 - f2),h)) . (k + 1) is
Function of
REAL,
REAL
by Th19;
A8:
(cdif (f2,h)) . (k + 1) is
Function of
REAL,
REAL
by Th19;
A9:
(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 A7, 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 A6
.=
((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 A9, Th5
.=
((cD (((cdif (f1,h)) . (k + 1)),h)) . x) - ((cD (((cdif (f2,h)) . (k + 1)),h)) . x)
by A8, 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;
for n being Nat holds S1[n]
from NAT_1:sch 2(A1, A4);
hence
((cdif ((f1 - f2),h)) . (n + 1)) . x = (((cdif (f1,h)) . (n + 1)) . x) - (((cdif (f2,h)) . (n + 1)) . x)
; verum