let F, G be Field; for V being VectSp of F
for W being VectSp of G
for f1, f2 being Function of V,W
for x, h being Element of V
for n being Nat holds ((cdif ((f1 + f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) + (((cdif (f2,h)) . (n + 1)) /. x)
let V be VectSp of F; for W being VectSp of G
for f1, f2 being Function of V,W
for x, h being Element of V
for n being Nat holds ((cdif ((f1 + f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) + (((cdif (f2,h)) . (n + 1)) /. x)
let W be VectSp of G; for f1, f2 being Function of V,W
for x, h being Element of V
for n being Nat 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 V,W; for x, h being Element of V
for n being Nat holds ((cdif ((f1 + f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) + (((cdif (f2,h)) . (n + 1)) /. x)
let x, h be Element of V; for n being Nat holds ((cdif ((f1 + f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) + (((cdif (f2,h)) . (n + 1)) /. x)
let n be Nat; ((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 Element of V holds ((cdif ((f1 + f2),h)) . ($1 + 1)) /. x = (((cdif (f1,h)) . ($1 + 1)) /. x) + (((cdif (f2,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 ((f1 + f2),h)) . (k + 1)) /. x = (((cdif (f1,h)) . (k + 1)) /. x) + (((cdif (f2,h)) . (k + 1)) /. x)
;
S1[k + 1]
let x be
Element of
V;
((cdif ((f1 + f2),h)) . ((k + 1) + 1)) /. x = (((cdif (f1,h)) . ((k + 1) + 1)) /. x) + (((cdif (f2,h)) . ((k + 1) + 1)) /. x)
A4:
(cdif ((f1 + f2),h)) . (k + 1) is
Function of
V,
W
by Th19;
A5:
(cdif (f2,h)) . (k + 1) is
Function of
V,
W
by Th19;
A6:
(cdif (f1,h)) . (k + 1) is
Function of
V,
W
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 + (((2 * (1. F)) ") * h))) - (((cdif ((f1 + f2),h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))
by A4, Th5
.=
((((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) + (((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h)))) - (((cdif ((f1 + f2),h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))
by A2
.=
((((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) + (((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h)))) - ((((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) + (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))))
by A2
.=
(((((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) + (((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h)))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))) - (((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))
by RLVECT_1:27
.=
((((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) + ((((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))))) - (((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))
by RLVECT_1:28
.=
((((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))) + ((((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))))
by RLVECT_1:28
.=
((cD (((cdif (f1,h)) . (k + 1)),h)) /. x) + ((((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))))
by A6, Th5
.=
((cD (((cdif (f1,h)) . (k + 1)),h)) /. x) + ((cD (((cdif (f2,h)) . (k + 1)),h)) /. x)
by A5, 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;
B1: dom (f1 + f2) =
(dom f1) /\ (dom f2)
by VFUNCT_1:def 1
.=
the carrier of V /\ (dom f2)
by FUNCT_2:def 1
.=
the carrier of V /\ the carrier of V
by FUNCT_2:def 1
.=
the carrier of V
;
A7:
S1[ 0 ]
proof
let x be
Element of
V;
((cdif ((f1 + f2),h)) . (0 + 1)) /. x = (((cdif (f1,h)) . (0 + 1)) /. x) + (((cdif (f2,h)) . (0 + 1)) /. x)
reconsider xx =
x,
hp =
((2 * (1. F)) ") * h as
Element of
V ;
((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 + (((2 * (1. F)) ") * h))) - ((f1 + f2) /. (x - (((2 * (1. F)) ") * h)))
by Th5
.=
((f1 /. (xx + (((2 * (1. F)) ") * h))) + (f2 /. (xx + hp))) - ((f1 + f2) /. (xx - hp))
by B1, VFUNCT_1:def 1
.=
((f1 /. (x + (((2 * (1. F)) ") * h))) + (f2 /. (x + hp))) - ((f1 /. (x - (((2 * (1. F)) ") * h))) + (f2 /. (x - hp)))
by B1, VFUNCT_1:def 1
.=
(((f1 /. (x + (((2 * (1. F)) ") * h))) + (f2 /. (x + hp))) - (f1 /. (x - (((2 * (1. F)) ") * h)))) - (f2 /. (x - hp))
by RLVECT_1:27
.=
((f2 /. (x + hp)) + ((f1 /. (x + (((2 * (1. F)) ") * h))) - (f1 /. (x - (((2 * (1. F)) ") * h))))) - (f2 /. (x - hp))
by RLVECT_1:28
.=
((f1 /. (x + (((2 * (1. F)) ") * h))) - (f1 /. (x - (((2 * (1. F)) ") * h)))) + ((f2 /. (x + (((2 * (1. F)) ") * h))) - (f2 /. (x - (((2 * (1. F)) ") * h))))
by RLVECT_1:28
.=
((cD (f1,h)) /. x) + ((f2 /. (x + (((2 * (1. F)) ") * h))) - (f2 /. (x - (((2 * (1. F)) ") * h))))
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;
for n being Nat holds S1[n]
from NAT_1:sch 2(A7, A1);
hence
((cdif ((f1 + f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) + (((cdif (f2,h)) . (n + 1)) /. x)
; verum