let F, G be Field; for V being VectSp of F
for W being VectSp of G
for f being Function of V,W
for x, h being Element of V
for r being Element of G
for n being Nat holds ((cdif ((r (#) f),h)) . (n + 1)) /. x = r * (((cdif (f,h)) . (n + 1)) /. x)
let V be VectSp of F; for W being VectSp of G
for f being Function of V,W
for x, h being Element of V
for r being Element of G
for n being Nat holds ((cdif ((r (#) f),h)) . (n + 1)) /. x = r * (((cdif (f,h)) . (n + 1)) /. x)
let W be VectSp of G; for f being Function of V,W
for x, h being Element of V
for r being Element of G
for n being Nat holds ((cdif ((r (#) f),h)) . (n + 1)) /. x = r * (((cdif (f,h)) . (n + 1)) /. x)
let f be Function of V,W; for x, h being Element of V
for r being Element of G
for n being Nat holds ((cdif ((r (#) f),h)) . (n + 1)) /. x = r * (((cdif (f,h)) . (n + 1)) /. x)
let x, h be Element of V; for r being Element of G
for n being Nat holds ((cdif ((r (#) f),h)) . (n + 1)) /. x = r * (((cdif (f,h)) . (n + 1)) /. x)
let r be Element of G; for n being Nat holds ((cdif ((r (#) f),h)) . (n + 1)) /. x = r * (((cdif (f,h)) . (n + 1)) /. x)
let n be Nat; ((cdif ((r (#) f),h)) . (n + 1)) /. x = r * (((cdif (f,h)) . (n + 1)) /. x)
defpred S1[ Nat] means for x being Element of V 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
Element of
V holds
((cdif ((r (#) f),h)) . (k + 1)) /. x = r * (((cdif (f,h)) . (k + 1)) /. x)
;
S1[k + 1]
let x be
Element of
V;
((cdif ((r (#) f),h)) . ((k + 1) + 1)) /. x = r * (((cdif (f,h)) . ((k + 1) + 1)) /. x)
A3:
(
((cdif ((r (#) f),h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)) = r * (((cdif (f,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) &
((cdif ((r (#) f),h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h)) = r * (((cdif (f,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) )
by A2;
A4:
(cdif ((r (#) f),h)) . (k + 1) is
Function of
V,
W
by Th19;
A5:
(cdif (f,h)) . (k + 1) is
Function of
V,
W
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 + (((2 * (1. F)) ") * h))) - (((cdif ((r (#) f),h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))
by A4, Th5
.=
r * ((((cdif (f,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))))
by VECTSP_1:23, 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
Element of
V;
((cdif ((r (#) f),h)) . (0 + 1)) /. x = r * (((cdif (f,h)) . (0 + 1)) /. x)
x + (((2 * (1. F)) ") * h) in the
carrier of
V
;
then A7:
x + (((2 * (1. F)) ") * h) in dom (r (#) f)
by FUNCT_2:def 1;
x - (((2 * (1. F)) ") * h) in the
carrier of
V
;
then A8:
x - (((2 * (1. F)) ") * h) 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 + (((2 * (1. F)) ") * h))) - ((r (#) f) /. (x - (((2 * (1. F)) ") * h)))
by Th5
.=
(r * (f /. (x + (((2 * (1. F)) ") * h)))) - ((r (#) f) /. (x - (((2 * (1. F)) ") * h)))
by A7, Def4X
.=
(r * (f /. (x + (((2 * (1. F)) ") * h)))) - (r * (f /. (x - (((2 * (1. F)) ") * h))))
by A8, Def4X
.=
r * ((f /. (x + (((2 * (1. F)) ") * h))) - (f /. (x - (((2 * (1. F)) ") * h))))
by VECTSP_1:23
.=
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