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 h being Element of V
for n being Nat st f is constant holds
for x being Element of V holds ((cdif (f,h)) . (n + 1)) /. x = 0. W
let V be VectSp of F; for W being VectSp of G
for f being Function of V,W
for h being Element of V
for n being Nat st f is constant holds
for x being Element of V holds ((cdif (f,h)) . (n + 1)) /. x = 0. W
let W be VectSp of G; for f being Function of V,W
for h being Element of V
for n being Nat st f is constant holds
for x being Element of V holds ((cdif (f,h)) . (n + 1)) /. x = 0. W
let f be Function of V,W; for h being Element of V
for n being Nat st f is constant holds
for x being Element of V holds ((cdif (f,h)) . (n + 1)) /. x = 0. W
let h be Element of V; for n being Nat st f is constant holds
for x being Element of V holds ((cdif (f,h)) . (n + 1)) /. x = 0. W
let n be Nat; ( f is constant implies for x being Element of V holds ((cdif (f,h)) . (n + 1)) /. x = 0. W )
defpred S1[ Nat] means for x being Element of V holds ((cdif (f,h)) . ($1 + 1)) /. x = 0. W;
assume A1:
f is constant
; for x being Element of V holds ((cdif (f,h)) . (n + 1)) /. x = 0. W
A2:
for x being Element of V holds (f /. (x + (((2 * (1. F)) ") * h))) - (f /. (x - (((2 * (1. F)) ") * h))) = 0. W
A4:
S1[ 0 ]
A5:
for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be
Nat;
( S1[k] implies S1[k + 1] )
assume A6:
for
x being
Element of
V holds
((cdif (f,h)) . (k + 1)) /. x = 0. W
;
S1[k + 1]
let x be
Element of
V;
((cdif (f,h)) . ((k + 1) + 1)) /. x = 0. W
A8:
(cdif (f,h)) . (k + 1) is
Function of
V,
W
by Th19;
((cdif (f,h)) . (k + 2)) /. x =
((cdif (f,h)) . ((k + 1) + 1)) /. x
.=
(cD (((cdif (f,h)) . (k + 1)),h)) /. x
by Def8
.=
(((cdif (f,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))
by A8, Th5
.=
(((cdif (f,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (0. W)
by A6
.=
((cdif (f,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))
by RLVECT_1:13
.=
0. W
by A6
;
hence
((cdif (f,h)) . ((k + 1) + 1)) /. x = 0. W
;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(A4, A5);
hence
for x being Element of V holds ((cdif (f,h)) . (n + 1)) /. x = 0. W
; verum