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 ((bdif ((f1 - f2),h)) . (n + 1)) /. x = (((bdif (f1,h)) . (n + 1)) /. x) - (((bdif (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 ((bdif ((f1 - f2),h)) . (n + 1)) /. x = (((bdif (f1,h)) . (n + 1)) /. x) - (((bdif (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 ((bdif ((f1 - f2),h)) . (n + 1)) /. x = (((bdif (f1,h)) . (n + 1)) /. x) - (((bdif (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 ((bdif ((f1 - f2),h)) . (n + 1)) /. x = (((bdif (f1,h)) . (n + 1)) /. x) - (((bdif (f2,h)) . (n + 1)) /. x)
let x, h be Element of V; for n being Nat holds ((bdif ((f1 - f2),h)) . (n + 1)) /. x = (((bdif (f1,h)) . (n + 1)) /. x) - (((bdif (f2,h)) . (n + 1)) /. x)
let n be Nat; ((bdif ((f1 - f2),h)) . (n + 1)) /. x = (((bdif (f1,h)) . (n + 1)) /. x) - (((bdif (f2,h)) . (n + 1)) /. x)
defpred S1[ Nat] means for x being Element of V holds ((bdif ((f1 - f2),h)) . ($1 + 1)) /. x = (((bdif (f1,h)) . ($1 + 1)) /. x) - (((bdif (f2,h)) . ($1 + 1)) /. x);
A1:
S1[ 0 ]
proof
let x be
Element of
V;
((bdif ((f1 - f2),h)) . (0 + 1)) /. x = (((bdif (f1,h)) . (0 + 1)) /. x) - (((bdif (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;
((bdif ((f1 - f2),h)) . (0 + 1)) /. x =
(bD (((bdif ((f1 - f2),h)) . 0),h)) /. x
by Def7
.=
(bD ((f1 - f2),h)) /. x
by Def7
.=
((f1 - f2) /. x) - ((f1 - f2) /. (x - h))
by Th4
.=
((f1 /. x) - (f2 /. x)) - ((f1 - f2) /. (x - h))
by A2, VFUNCT_1:def 2
.=
((f1 /. x) - (f2 /. x)) - ((f1 /. (x - h)) - (f2 /. (x - h)))
by A3, VFUNCT_1:def 2
.=
(((f1 /. x) - (f2 /. x)) - (f1 /. (x - h))) + (f2 /. (x - h))
by RLVECT_1:29
.=
((f1 /. x) - ((f1 /. (x - h)) + (f2 /. x))) + (f2 /. (x - h))
by RLVECT_1:27
.=
(((f1 /. x) - (f1 /. (x - h))) - (f2 /. x)) + (f2 /. (x - h))
by RLVECT_1:27
.=
((f1 /. x) - (f1 /. (x - h))) - ((f2 /. x) - (f2 /. (x - h)))
by RLVECT_1:29
.=
((bD (f1,h)) /. x) - ((f2 /. x) - (f2 /. (x - h)))
by Th4
.=
((bD (f1,h)) /. x) - ((bD (f2,h)) /. x)
by Th4
.=
((bD (((bdif (f1,h)) . 0),h)) /. x) - ((bD (f2,h)) /. x)
by Def7
.=
((bD (((bdif (f1,h)) . 0),h)) /. x) - ((bD (((bdif (f2,h)) . 0),h)) /. x)
by Def7
.=
(((bdif (f1,h)) . (0 + 1)) /. x) - ((bD (((bdif (f2,h)) . 0),h)) /. x)
by Def7
.=
(((bdif (f1,h)) . (0 + 1)) /. x) - (((bdif (f2,h)) . (0 + 1)) /. x)
by Def7
;
hence
((bdif ((f1 - f2),h)) . (0 + 1)) /. x = (((bdif (f1,h)) . (0 + 1)) /. x) - (((bdif (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
((bdif ((f1 - f2),h)) . (k + 1)) /. x = (((bdif (f1,h)) . (k + 1)) /. x) - (((bdif (f2,h)) . (k + 1)) /. x)
;
S1[k + 1]
let x be
Element of
V;
((bdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x = (((bdif (f1,h)) . ((k + 1) + 1)) /. x) - (((bdif (f2,h)) . ((k + 1) + 1)) /. x)
A6:
(
((bdif ((f1 - f2),h)) . (k + 1)) /. x = (((bdif (f1,h)) . (k + 1)) /. x) - (((bdif (f2,h)) . (k + 1)) /. x) &
((bdif ((f1 - f2),h)) . (k + 1)) /. (x - h) = (((bdif (f1,h)) . (k + 1)) /. (x - h)) - (((bdif (f2,h)) . (k + 1)) /. (x - h)) )
by A5;
A7:
(bdif ((f1 - f2),h)) . (k + 1) is
Function of
V,
W
by Th12;
A8:
(bdif (f2,h)) . (k + 1) is
Function of
V,
W
by Th12;
A9:
(bdif (f1,h)) . (k + 1) is
Function of
V,
W
by Th12;
((bdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x =
(bD (((bdif ((f1 - f2),h)) . (k + 1)),h)) /. x
by Def7
.=
(((bdif ((f1 - f2),h)) . (k + 1)) /. x) - (((bdif ((f1 - f2),h)) . (k + 1)) /. (x - h))
by A7, Th4
.=
(((((bdif (f1,h)) . (k + 1)) /. x) - (((bdif (f2,h)) . (k + 1)) /. x)) - (((bdif (f1,h)) . (k + 1)) /. (x - h))) + (((bdif (f2,h)) . (k + 1)) /. (x - h))
by RLVECT_1:29, A6
.=
((((bdif (f1,h)) . (k + 1)) /. x) - ((((bdif (f1,h)) . (k + 1)) /. (x - h)) + (((bdif (f2,h)) . (k + 1)) /. x))) + (((bdif (f2,h)) . (k + 1)) /. (x - h))
by RLVECT_1:27
.=
(((((bdif (f1,h)) . (k + 1)) /. x) - (((bdif (f1,h)) . (k + 1)) /. (x - h))) - (((bdif (f2,h)) . (k + 1)) /. x)) + (((bdif (f2,h)) . (k + 1)) /. (x - h))
by RLVECT_1:27
.=
((((bdif (f1,h)) . (k + 1)) /. x) - (((bdif (f1,h)) . (k + 1)) /. (x - h))) - ((((bdif (f2,h)) . (k + 1)) /. x) - (((bdif (f2,h)) . (k + 1)) /. (x - h)))
by RLVECT_1:29
.=
((bD (((bdif (f1,h)) . (k + 1)),h)) /. x) - ((((bdif (f2,h)) . (k + 1)) /. x) - (((bdif (f2,h)) . (k + 1)) /. (x - h)))
by A9, Th4
.=
((bD (((bdif (f1,h)) . (k + 1)),h)) /. x) - ((bD (((bdif (f2,h)) . (k + 1)),h)) /. x)
by A8, Th4
.=
(((bdif (f1,h)) . ((k + 1) + 1)) /. x) - ((bD (((bdif (f2,h)) . (k + 1)),h)) /. x)
by Def7
.=
(((bdif (f1,h)) . ((k + 1) + 1)) /. x) - (((bdif (f2,h)) . ((k + 1) + 1)) /. x)
by Def7
;
hence
((bdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x = (((bdif (f1,h)) . ((k + 1) + 1)) /. x) - (((bdif (f2,h)) . ((k + 1) + 1)) /. x)
;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(A1, A4);
hence
((bdif ((f1 - f2),h)) . (n + 1)) /. x = (((bdif (f1,h)) . (n + 1)) /. x) - (((bdif (f2,h)) . (n + 1)) /. x)
; verum