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 ((fdif (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 ((fdif (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 ((fdif (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 ((fdif (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 ((fdif (f,h)) . (n + 1)) /. x = 0. W
let n be Nat; ( f is constant implies for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W )
assume A1:
f is constant
; for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W
A2:
for x being Element of V holds (f /. (x + h)) - (f /. x) = 0. W
for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W
proof
defpred S1[
Nat]
means for
x being
Element of
V holds
((fdif (f,h)) . ($1 + 1)) /. x = 0. W;
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
Element of
V holds
((fdif (f,h)) . (k + 1)) /. x = 0. W
;
S1[k + 1]
let x be
Element of
V;
((fdif (f,h)) . ((k + 1) + 1)) /. x = 0. W
A6:
((fdif (f,h)) . (k + 1)) /. (x + h) = 0. W
by A5;
reconsider fdk =
(fdif (f,h)) . (k + 1) as
Function of
V,
W by Th2;
((fdif (f,h)) . (k + 2)) /. x =
((fdif (f,h)) . ((k + 1) + 1)) /. x
.=
(fD (((fdif (f,h)) . (k + 1)),h)) /. x
by Def6
.=
(fdk /. (x + h)) - (fdk /. x)
by Th3
.=
(0. W) - (0. W)
by A5, A6
.=
0. W
by RLVECT_1:15
;
hence
((fdif (f,h)) . ((k + 1) + 1)) /. x = 0. W
;
verum
end;
A8:
S1[
0 ]
for
n being
Nat holds
S1[
n]
from NAT_1:sch 2(A8, A4);
hence
for
x being
Element of
V holds
((fdif (f,h)) . (n + 1)) /. x = 0. W
;
verum
end;
hence
for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W
; verum