let n be Element of NAT ; :: thesis: for r, h, x being Real
for f being Function of REAL , REAL holds ((cdif (r (#) f),h) . (n + 1)) . x = r * (((cdif f,h) . (n + 1)) . x)
let r, h, x be Real; :: thesis: for f being Function of REAL , REAL holds ((cdif (r (#) f),h) . (n + 1)) . x = r * (((cdif f,h) . (n + 1)) . x)
let f be Function of REAL , REAL ; :: thesis: ((cdif (r (#) f),h) . (n + 1)) . x = r * (((cdif f,h) . (n + 1)) . x)
defpred S1[ Element of NAT ] means for x being Real holds ((cdif (r (#) f),h) . ($1 + 1)) . x = r * (((cdif f,h) . ($1 + 1)) . x);
A1:
S1[ 0 ]
proof
let x be
Real;
:: thesis: ((cdif (r (#) f),h) . (0 + 1)) . x = r * (((cdif f,h) . (0 + 1)) . x)
x + (h / 2) in REAL
;
then A2:
x + (h / 2) in dom (r (#) f)
by FUNCT_2:def 1;
x - (h / 2) in REAL
;
then A3:
x - (h / 2) in dom (r (#) f)
by FUNCT_2:def 1;
((cdif (r (#) f),h) . (0 + 1)) . x =
(cD ((cdif (r (#) f),h) . 0 ),h) . x
by Def8
.=
(cD (r (#) f),h) . x
by Def8
.=
((r (#) f) . (x + (h / 2))) - ((r (#) f) . (x - (h / 2)))
by Th5
.=
(r * (f . (x + (h / 2)))) - ((r (#) f) . (x - (h / 2)))
by A2, VALUED_1:def 5
.=
(r * (f . (x + (h / 2)))) - (r * (f . (x - (h / 2))))
by A3, VALUED_1:def 5
.=
r * ((f . (x + (h / 2))) - (f . (x - (h / 2))))
.=
r * ((cD f,h) . x)
by Th5
.=
r * ((cD ((cdif f,h) . 0 ),h) . x)
by Def8
.=
r * (((cdif f,h) . (0 + 1)) . x)
by Def8
;
hence
((cdif (r (#) f),h) . (0 + 1)) . x = r * (((cdif f,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 (r (#) f),h) . (k + 1)) . x = r * (((cdif f,h) . (k + 1)) . x)
;
:: thesis: S1[k + 1]
let x be
Real;
:: thesis: ((cdif (r (#) f),h) . ((k + 1) + 1)) . x = r * (((cdif f,h) . ((k + 1) + 1)) . x)
A6:
((cdif (r (#) f),h) . (k + 1)) . (x - (h / 2)) = r * (((cdif f,h) . (k + 1)) . (x - (h / 2)))
by A5;
A7:
((cdif (r (#) f),h) . (k + 1)) . (x + (h / 2)) = r * (((cdif f,h) . (k + 1)) . (x + (h / 2)))
by A5;
A8:
(cdif (r (#) f),h) . (k + 1) is
Function of
REAL ,
REAL
by Th19;
A9:
(cdif f,h) . (k + 1) is
Function of
REAL ,
REAL
by Th19;
((cdif (r (#) f),h) . ((k + 1) + 1)) . x =
(cD ((cdif (r (#) f),h) . (k + 1)),h) . x
by Def8
.=
(((cdif (r (#) f),h) . (k + 1)) . (x + (h / 2))) - (((cdif (r (#) f),h) . (k + 1)) . (x - (h / 2)))
by A8, Th5
.=
r * ((((cdif f,h) . (k + 1)) . (x + (h / 2))) - (((cdif f,h) . (k + 1)) . (x - (h / 2))))
by A6, A7
.=
r * ((cD ((cdif f,h) . (k + 1)),h) . x)
by A9, Th5
.=
r * (((cdif f,h) . ((k + 1) + 1)) . x)
by Def8
;
hence
((cdif (r (#) f),h) . ((k + 1) + 1)) . x = r * (((cdif f,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 (r (#) f),h) . (n + 1)) . x = r * (((cdif f,h) . (n + 1)) . x)
; :: thesis: verum