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 ((fdif ((f1 - f2),h)) . (n + 1)) /. x = (((fdif (f1,h)) . (n + 1)) /. x) - (((fdif (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 ((fdif ((f1 - f2),h)) . (n + 1)) /. x = (((fdif (f1,h)) . (n + 1)) /. x) - (((fdif (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 ((fdif ((f1 - f2),h)) . (n + 1)) /. x = (((fdif (f1,h)) . (n + 1)) /. x) - (((fdif (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 ((fdif ((f1 - f2),h)) . (n + 1)) /. x = (((fdif (f1,h)) . (n + 1)) /. x) - (((fdif (f2,h)) . (n + 1)) /. x)
let x, h be Element of V; for n being Nat holds ((fdif ((f1 - f2),h)) . (n + 1)) /. x = (((fdif (f1,h)) . (n + 1)) /. x) - (((fdif (f2,h)) . (n + 1)) /. x)
let n be Nat; ((fdif ((f1 - f2),h)) . (n + 1)) /. x = (((fdif (f1,h)) . (n + 1)) /. x) - (((fdif (f2,h)) . (n + 1)) /. x)
defpred S1[ Nat] means for x being Element of V holds ((fdif ((f1 - f2),h)) . ($1 + 1)) /. x = (((fdif (f1,h)) . ($1 + 1)) /. x) - (((fdif (f2,h)) . ($1 + 1)) /. x);
A1:
S1[ 0 ]
proof
let x be
Element of
V;
((fdif ((f1 - f2),h)) . (0 + 1)) /. x = (((fdif (f1,h)) . (0 + 1)) /. x) - (((fdif (f2,h)) . (0 + 1)) /. x)
x in the
carrier of
V
;
then
(
x in dom f1 &
x in dom f2 )
by FUNCT_2:def 1;
then
x in (dom f1) /\ (dom f2)
by XBOOLE_0:def 4;
then A2:
x in dom (f1 - f2)
by VFUNCT_1:def 2;
x + h in the
carrier of
V
;
then
(
x + h in dom f1 &
x + h in dom f2 )
by FUNCT_2:def 1;
then
x + h in (dom f1) /\ (dom f2)
by XBOOLE_0:def 4;
then A3:
x + h in dom (f1 - f2)
by VFUNCT_1:def 2;
((fdif ((f1 - f2),h)) . (0 + 1)) /. x =
(fD (((fdif ((f1 - f2),h)) . 0),h)) /. x
by Def6
.=
(fD ((f1 - f2),h)) /. x
by Def6
.=
((f1 - f2) /. (x + h)) - ((f1 - f2) /. x)
by Th3
.=
((f1 /. (x + h)) - (f2 /. (x + h))) - ((f1 - f2) /. x)
by A3, VFUNCT_1:def 2
.=
((f1 /. (x + h)) - (f2 /. (x + h))) - ((f1 /. x) - (f2 /. x))
by A2, VFUNCT_1:def 2
.=
(((f1 /. (x + h)) - (f2 /. (x + h))) - (f1 /. x)) + (f2 /. x)
by RLVECT_1:29
.=
((f1 /. (x + h)) - ((f1 /. x) + (f2 /. (x + h)))) + (f2 /. x)
by RLVECT_1:27
.=
(((f1 /. (x + h)) - (f1 /. x)) - (f2 /. (x + h))) + (f2 /. x)
by RLVECT_1:27
.=
(((fD (f1,h)) /. x) - (f2 /. (x + h))) + (f2 /. x)
by Th3
.=
((fD (f1,h)) . x) - ((f2 /. (x + h)) - (f2 /. x))
by RLVECT_1:29
.=
((fD (f1,h)) /. x) - ((fD (f2,h)) /. x)
by Th3
.=
((fD (((fdif (f1,h)) . 0),h)) /. x) - ((fD (f2,h)) /. x)
by Def6
.=
((fD (((fdif (f1,h)) . 0),h)) /. x) - ((fD (((fdif (f2,h)) . 0),h)) /. x)
by Def6
.=
(((fdif (f1,h)) . (0 + 1)) /. x) - ((fD (((fdif (f2,h)) . 0),h)) /. x)
by Def6
.=
(((fdif (f1,h)) . (0 + 1)) /. x) - (((fdif (f2,h)) . (0 + 1)) /. x)
by Def6
;
hence
((fdif ((f1 - f2),h)) . (0 + 1)) /. x = (((fdif (f1,h)) . (0 + 1)) /. x) - (((fdif (f2,h)) . (0 + 1)) /. x)
;
verum
end;
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 ((f1 - f2),h)) . (k + 1)) /. x = (((fdif (f1,h)) . (k + 1)) /. x) - (((fdif (f2,h)) . (k + 1)) /. x)
;
S1[k + 1]
let x be
Element of
V;
((fdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x = (((fdif (f1,h)) . ((k + 1) + 1)) /. x) - (((fdif (f2,h)) . ((k + 1) + 1)) /. x)
A6:
(
((fdif ((f1 - f2),h)) . (k + 1)) /. x = (((fdif (f1,h)) . (k + 1)) /. x) - (((fdif (f2,h)) . (k + 1)) /. x) &
((fdif ((f1 - f2),h)) . (k + 1)) /. (x + h) = (((fdif (f1,h)) . (k + 1)) /. (x + h)) - (((fdif (f2,h)) . (k + 1)) /. (x + h)) )
by A5;
reconsider fd12k1 =
(fdif ((f1 - f2),h)) . (k + 1) as
Function of
V,
W by Th2;
reconsider fd2k =
(fdif (f2,h)) . (k + 1) as
Function of
V,
W by Th2;
reconsider fd1k =
(fdif (f1,h)) . (k + 1) as
Function of
V,
W by Th2;
reconsider fdiff12 =
(fdif ((f1 - f2),h)) . (k + 1) as
Function of
V,
W by Th2;
reconsider fdiff2 =
(fdif (f2,h)) . (k + 1) as
Function of
V,
W by Th2;
reconsider fdiff1 =
(fdif (f1,h)) . (k + 1) as
Function of
V,
W by Th2;
A12:
(fD (((fdif (f1,h)) . (k + 1)),h)) /. x =
(fD (fdiff1,h)) /. x
.=
(((fdif (f1,h)) . (k + 1)) /. (x + h)) - (((fdif (f1,h)) . (k + 1)) /. x)
by Th3
;
A13:
(fD (((fdif (f2,h)) . (k + 1)),h)) /. x =
(fD (fdiff2,h)) /. x
.=
(((fdif (f2,h)) . (k + 1)) /. (x + h)) - (((fdif (f2,h)) . (k + 1)) /. x)
by Th3
;
((fdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x =
(fD (((fdif ((f1 - f2),h)) . (k + 1)),h)) /. x
by Def6
.=
(fd12k1 /. (x + h)) - (fd12k1 /. x)
by Th3
.=
(((((fdif (f1,h)) . (k + 1)) /. (x + h)) + (- (((fdif (f2,h)) . (k + 1)) /. (x + h)))) - (((fdif (f1,h)) . (k + 1)) /. x)) + (((fdif (f2,h)) . (k + 1)) /. x)
by RLVECT_1:29, A6
.=
((((fdif (f1,h)) . (k + 1)) /. (x + h)) + ((- (((fdif (f2,h)) . (k + 1)) /. (x + h))) + (- (((fdif (f1,h)) . (k + 1)) /. x)))) + (((fdif (f2,h)) . (k + 1)) /. x)
by RLVECT_1:def 3
.=
(((((fdif (f1,h)) . (k + 1)) /. (x + h)) + (- (((fdif (f1,h)) . (k + 1)) /. x))) - (((fdif (f2,h)) . (k + 1)) /. (x + h))) + (((fdif (f2,h)) . (k + 1)) /. x)
by RLVECT_1:def 3
.=
((fD (((fdif (f1,h)) . (k + 1)),h)) /. x) - ((((fdif (f2,h)) . (k + 1)) /. (x + h)) - (((fdif (f2,h)) . (k + 1)) /. x))
by A12, RLVECT_1:29
.=
(((fdif (f1,h)) . ((k + 1) + 1)) /. x) - ((fD (((fdif (f2,h)) . (k + 1)),h)) /. x)
by Def6, A13
.=
(((fdif (f1,h)) . ((k + 1) + 1)) /. x) - (((fdif (f2,h)) . ((k + 1) + 1)) /. x)
by Def6
;
hence
((fdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x = (((fdif (f1,h)) . ((k + 1) + 1)) /. x) - (((fdif (f2,h)) . ((k + 1) + 1)) /. x)
;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(A1, A4);
hence
((fdif ((f1 - f2),h)) . (n + 1)) /. x = (((fdif (f1,h)) . (n + 1)) /. x) - (((fdif (f2,h)) . (n + 1)) /. x)
; verum