let n be Nat; :: thesis: for h being Real

for f being Function of REAL,REAL st f is constant holds

for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0

let h be Real; :: thesis: for f being Function of REAL,REAL st f is constant holds

for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0

let f be Function of REAL,REAL; :: thesis: ( f is constant implies for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0 )

defpred S_{1}[ Nat] means for x being Real holds ((cdif (f,h)) . ($1 + 1)) . x = 0 ;

assume A1: f is constant ; :: thesis: for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0

A2: for x being Real holds (f . (x + (h / 2))) - (f . (x - (h / 2))) = 0_{1}[ 0 ]
_{1}[k] holds

S_{1}[k + 1]
_{1}[n]
from NAT_1:sch 2(A4, A5);

hence for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0 ; :: thesis: verum

for f being Function of REAL,REAL st f is constant holds

for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0

let h be Real; :: thesis: for f being Function of REAL,REAL st f is constant holds

for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0

let f be Function of REAL,REAL; :: thesis: ( f is constant implies for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0 )

defpred S

assume A1: f is constant ; :: thesis: for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0

A2: for x being Real holds (f . (x + (h / 2))) - (f . (x - (h / 2))) = 0

proof

A4:
S
let x be Real; :: thesis: (f . (x + (h / 2))) - (f . (x - (h / 2))) = 0

x - (h / 2) in REAL by XREAL_0:def 1;

then A3: x - (h / 2) in dom f by FUNCT_2:def 1;

x + (h / 2) in REAL by XREAL_0:def 1;

then x + (h / 2) in dom f by FUNCT_2:def 1;

then f . (x + (h / 2)) = f . (x - (h / 2)) by A1, A3, FUNCT_1:def 10;

hence (f . (x + (h / 2))) - (f . (x - (h / 2))) = 0 ; :: thesis: verum

end;x - (h / 2) in REAL by XREAL_0:def 1;

then A3: x - (h / 2) in dom f by FUNCT_2:def 1;

x + (h / 2) in REAL by XREAL_0:def 1;

then x + (h / 2) in dom f by FUNCT_2:def 1;

then f . (x + (h / 2)) = f . (x - (h / 2)) by A1, A3, FUNCT_1:def 10;

hence (f . (x + (h / 2))) - (f . (x - (h / 2))) = 0 ; :: thesis: verum

proof

A5:
for k being Nat st S
let x be Real; :: thesis: ((cdif (f,h)) . (0 + 1)) . x = 0

thus ((cdif (f,h)) . (0 + 1)) . x = (cD (((cdif (f,h)) . 0),h)) . x by Def8

.= (cD (f,h)) . x by Def8

.= (f . (x + (h / 2))) - (f . (x - (h / 2))) by Th5

.= 0 by A2 ; :: thesis: verum

end;thus ((cdif (f,h)) . (0 + 1)) . x = (cD (((cdif (f,h)) . 0),h)) . x by Def8

.= (cD (f,h)) . x by Def8

.= (f . (x + (h / 2))) - (f . (x - (h / 2))) by Th5

.= 0 by A2 ; :: thesis: verum

S

proof

for n being Nat holds S
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A6: for x being Real holds ((cdif (f,h)) . (k + 1)) . x = 0 ; :: thesis: S_{1}[k + 1]

let x be Real; :: thesis: ((cdif (f,h)) . ((k + 1) + 1)) . x = 0

A7: ((cdif (f,h)) . (k + 1)) . (x - (h / 2)) = 0 by A6;

A8: (cdif (f,h)) . (k + 1) is Function of REAL,REAL by Th19;

((cdif (f,h)) . (k + 2)) . x = ((cdif (f,h)) . ((k + 1) + 1)) . x

.= (cD (((cdif (f,h)) . (k + 1)),h)) . x by Def8

.= (((cdif (f,h)) . (k + 1)) . (x + (h / 2))) - (((cdif (f,h)) . (k + 1)) . (x - (h / 2))) by A8, Th5

.= 0 by A6, A7 ;

hence ((cdif (f,h)) . ((k + 1) + 1)) . x = 0 ; :: thesis: verum

end;assume A6: for x being Real holds ((cdif (f,h)) . (k + 1)) . x = 0 ; :: thesis: S

let x be Real; :: thesis: ((cdif (f,h)) . ((k + 1) + 1)) . x = 0

A7: ((cdif (f,h)) . (k + 1)) . (x - (h / 2)) = 0 by A6;

A8: (cdif (f,h)) . (k + 1) is Function of REAL,REAL by Th19;

((cdif (f,h)) . (k + 2)) . x = ((cdif (f,h)) . ((k + 1) + 1)) . x

.= (cD (((cdif (f,h)) . (k + 1)),h)) . x by Def8

.= (((cdif (f,h)) . (k + 1)) . (x + (h / 2))) - (((cdif (f,h)) . (k + 1)) . (x - (h / 2))) by A8, Th5

.= 0 by A6, A7 ;

hence ((cdif (f,h)) . ((k + 1) + 1)) . x = 0 ; :: thesis: verum

hence for x being Real holds ((cdif (f,h)) . (n + 1)) . x = 0 ; :: thesis: verum