let n be Element of NAT ; :: thesis: for h, x being Real
for f1, f2 being Function of REAL ,REAL holds ((cdif (f1 - f2),h) . (n + 1)) . x = (((cdif f1,h) . (n + 1)) . x) - (((cdif f2,h) . (n + 1)) . x)
let h, x be Real; :: thesis: for f1, f2 being Function of REAL ,REAL 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 REAL ,REAL ; :: thesis: ((cdif (f1 - f2),h) . (n + 1)) . x = (((cdif f1,h) . (n + 1)) . x) - (((cdif f2,h) . (n + 1)) . x)
defpred S1[ Element of NAT ] means for x being Real holds ((cdif (f1 - f2),h) . ($1 + 1)) . x = (((cdif f1,h) . ($1 + 1)) . x) - (((cdif f2,h) . ($1 + 1)) . x);
A1:
S1[ 0 ]
proof
let x be
Real;
:: thesis: ((cdif (f1 - f2),h) . (0 + 1)) . x = (((cdif f1,h) . (0 + 1)) . x) - (((cdif f2,h) . (0 + 1)) . x)
x - (h / 2) in REAL
;
then
(
x - (h / 2) in dom f1 &
x - (h / 2) in dom f2 )
by FUNCT_2:def 1;
then
x - (h / 2) in (dom f1) /\ (dom f2)
by XBOOLE_0:def 4;
then A2:
x - (h / 2) in dom (f1 - f2)
by VALUED_1:12;
x + (h / 2) in REAL
;
then
(
x + (h / 2) in dom f1 &
x + (h / 2) in dom f2 )
by FUNCT_2:def 1;
then
x + (h / 2) in (dom f1) /\ (dom f2)
by XBOOLE_0:def 4;
then A3:
x + (h / 2) in dom (f1 - f2)
by VALUED_1:12;
((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 + (h / 2))) - ((f1 - f2) . (x - (h / 2)))
by Th5
.=
((f1 . (x + (h / 2))) - (f2 . (x + (h / 2)))) - ((f1 - f2) . (x - (h / 2)))
by A3, VALUED_1:13
.=
((f1 . (x + (h / 2))) - (f2 . (x + (h / 2)))) - ((f1 . (x - (h / 2))) - (f2 . (x - (h / 2))))
by A2, VALUED_1:13
.=
((f1 . (x + (h / 2))) - (f1 . (x - (h / 2)))) - ((f2 . (x + (h / 2))) - (f2 . (x - (h / 2))))
.=
((cD f1,h) . x) - ((f2 . (x + (h / 2))) - (f2 . (x - (h / 2))))
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)
;
:: thesis: verum
end;
A4:
for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be
Element of
NAT ;
:: thesis: ( S1[k] implies S1[k + 1] )
assume A5:
for
x being
Real holds
((cdif (f1 - f2),h) . (k + 1)) . x = (((cdif f1,h) . (k + 1)) . x) - (((cdif f2,h) . (k + 1)) . x)
;
:: thesis: S1[k + 1]
let x be
Real;
:: thesis: ((cdif (f1 - f2),h) . ((k + 1) + 1)) . x = (((cdif f1,h) . ((k + 1) + 1)) . x) - (((cdif f2,h) . ((k + 1) + 1)) . x)
A6:
((cdif (f1 - f2),h) . (k + 1)) . (x - (h / 2)) = (((cdif f1,h) . (k + 1)) . (x - (h / 2))) - (((cdif f2,h) . (k + 1)) . (x - (h / 2)))
by A5;
A7:
((cdif (f1 - f2),h) . (k + 1)) . (x + (h / 2)) = (((cdif f1,h) . (k + 1)) . (x + (h / 2))) - (((cdif f2,h) . (k + 1)) . (x + (h / 2)))
by A5;
A8:
(cdif (f1 - f2),h) . (k + 1) is
Function of
REAL ,
REAL
by Th19;
A9:
(cdif f1,h) . (k + 1) is
Function of
REAL ,
REAL
by Th19;
A10:
(cdif f2,h) . (k + 1) is
Function of
REAL ,
REAL
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 + (h / 2))) - (((cdif (f1 - f2),h) . (k + 1)) . (x - (h / 2)))
by A8, Th5
.=
((((cdif f1,h) . (k + 1)) . (x + (h / 2))) - (((cdif f1,h) . (k + 1)) . (x - (h / 2)))) - ((((cdif f2,h) . (k + 1)) . (x + (h / 2))) - (((cdif f2,h) . (k + 1)) . (x - (h / 2))))
by A6, A7
.=
((cD ((cdif f1,h) . (k + 1)),h) . x) - ((((cdif f2,h) . (k + 1)) . (x + (h / 2))) - (((cdif f2,h) . (k + 1)) . (x - (h / 2))))
by A9, Th5
.=
((cD ((cdif f1,h) . (k + 1)),h) . x) - ((cD ((cdif f2,h) . (k + 1)),h) . x)
by A10, 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)
;
:: thesis: verum
end;
for n being Element of NAT holds S1[n]
from NAT_1:sch 1(A1, A4);
hence
((cdif (f1 - f2),h) . (n + 1)) . x = (((cdif f1,h) . (n + 1)) . x) - (((cdif f2,h) . (n + 1)) . x)
; :: thesis: verum