let n be Nat; for h, r, x being Real
for f being Function of REAL,REAL holds ((cdif ((r (#) f),h)) . (n + 1)) . x = r * (((cdif (f,h)) . (n + 1)) . x)
let h, r, x be Real; for f being Function of REAL,REAL holds ((cdif ((r (#) f),h)) . (n + 1)) . x = r * (((cdif (f,h)) . (n + 1)) . x)
let f be Function of REAL,REAL; ((cdif ((r (#) f),h)) . (n + 1)) . x = r * (((cdif (f,h)) . (n + 1)) . x)
defpred S1[ Nat] means for x being Real holds ((cdif ((r (#) f),h)) . ($1 + 1)) . x = r * (((cdif (f,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 ((r (#) f),h)) . (k + 1)) . x = r * (((cdif (f,h)) . (k + 1)) . x)
;
S1[k + 1]
let x be
Real;
((cdif ((r (#) f),h)) . ((k + 1) + 1)) . x = r * (((cdif (f,h)) . ((k + 1) + 1)) . x)
A3:
(
((cdif ((r (#) f),h)) . (k + 1)) . (x - (h / 2)) = r * (((cdif (f,h)) . (k + 1)) . (x - (h / 2))) &
((cdif ((r (#) f),h)) . (k + 1)) . (x + (h / 2)) = r * (((cdif (f,h)) . (k + 1)) . (x + (h / 2))) )
by A2;
A4:
(cdif ((r (#) f),h)) . (k + 1) is
Function of
REAL,
REAL
by Th19;
A5:
(cdif (f,h)) . (k + 1) is
Function of
REAL,
REAL
by Th19;
((cdif ((r (#) f),h)) . ((k + 1) + 1)) . x =
(cD (((cdif ((r (#) f),h)) . (k + 1)),h)) . x
by Def8
.=
(((cdif ((r (#) f),h)) . (k + 1)) . (x + (h / 2))) - (((cdif ((r (#) f),h)) . (k + 1)) . (x - (h / 2)))
by A4, Th5
.=
r * ((((cdif (f,h)) . (k + 1)) . (x + (h / 2))) - (((cdif (f,h)) . (k + 1)) . (x - (h / 2))))
by A3
.=
r * ((cD (((cdif (f,h)) . (k + 1)),h)) . x)
by A5, Th5
.=
r * (((cdif (f,h)) . ((k + 1) + 1)) . x)
by Def8
;
hence
((cdif ((r (#) f),h)) . ((k + 1) + 1)) . x = r * (((cdif (f,h)) . ((k + 1) + 1)) . x)
;
verum
end;
A6:
S1[ 0 ]
proof
let x be
Real;
((cdif ((r (#) f),h)) . (0 + 1)) . x = r * (((cdif (f,h)) . (0 + 1)) . x)
x + (h / 2) in REAL
by XREAL_0:def 1;
then A7:
x + (h / 2) in dom (r (#) f)
by FUNCT_2:def 1;
x - (h / 2) in REAL
by XREAL_0:def 1;
then A8:
x - (h / 2) in dom (r (#) f)
by FUNCT_2:def 1;
((cdif ((r (#) f),h)) . (0 + 1)) . x =
(cD (((cdif ((r (#) f),h)) . 0),h)) . x
by Def8
.=
(cD ((r (#) f),h)) . x
by Def8
.=
((r (#) f) . (x + (h / 2))) - ((r (#) f) . (x - (h / 2)))
by Th5
.=
(r * (f . (x + (h / 2)))) - ((r (#) f) . (x - (h / 2)))
by A7, VALUED_1:def 5
.=
(r * (f . (x + (h / 2)))) - (r * (f . (x - (h / 2))))
by A8, VALUED_1:def 5
.=
r * ((f . (x + (h / 2))) - (f . (x - (h / 2))))
.=
r * ((cD (f,h)) . x)
by Th5
.=
r * ((cD (((cdif (f,h)) . 0),h)) . x)
by Def8
.=
r * (((cdif (f,h)) . (0 + 1)) . x)
by Def8
;
hence
((cdif ((r (#) f),h)) . (0 + 1)) . x = r * (((cdif (f,h)) . (0 + 1)) . x)
;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(A6, A1);
hence
((cdif ((r (#) f),h)) . (n + 1)) . x = r * (((cdif (f,h)) . (n + 1)) . x)
; verum