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 ((bdif (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 ((bdif (f,h)) . (n + 1)) . x = 0

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

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

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

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

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

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

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

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

proof

for x being Real holds ((bdif (f,h)) . (n + 1)) . x = 0
let x be Real; :: thesis: (f . x) - (f . (x - h)) = 0

x - h in REAL by XREAL_0:def 1;

then A3: x - h in dom f by FUNCT_2:def 1;

x in REAL by XREAL_0:def 1;

then x in dom f by FUNCT_2:def 1;

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

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

end;x - h in REAL by XREAL_0:def 1;

then A3: x - h in dom f by FUNCT_2:def 1;

x in REAL by XREAL_0:def 1;

then x in dom f by FUNCT_2:def 1;

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

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

proof

hence
for x being Real holds ((bdif (f,h)) . (n + 1)) . x = 0
; :: thesis: verum
defpred S_{1}[ Nat] means for x being Real holds ((bdif (f,h)) . ($1 + 1)) . x = 0 ;

A4: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]
_{1}[ 0 ]
_{1}[n]
from NAT_1:sch 2(A8, A4);

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

end;A4: for k being Nat st S

S

proof

A8:
S
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A5: for x being Real holds ((bdif (f,h)) . (k + 1)) . x = 0 ; :: thesis: S_{1}[k + 1]

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

A6: ((bdif (f,h)) . (k + 1)) . (x - h) = 0 by A5;

A7: (bdif (f,h)) . (k + 1) is Function of REAL,REAL by Th12;

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

.= (bD (((bdif (f,h)) . (k + 1)),h)) . x by Def7

.= (((bdif (f,h)) . (k + 1)) . x) - (((bdif (f,h)) . (k + 1)) . (x - h)) by A7, Th4

.= 0 by A5, A6 ;

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

end;assume A5: for x being Real holds ((bdif (f,h)) . (k + 1)) . x = 0 ; :: thesis: S

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

A6: ((bdif (f,h)) . (k + 1)) . (x - h) = 0 by A5;

A7: (bdif (f,h)) . (k + 1) is Function of REAL,REAL by Th12;

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

.= (bD (((bdif (f,h)) . (k + 1)),h)) . x by Def7

.= (((bdif (f,h)) . (k + 1)) . x) - (((bdif (f,h)) . (k + 1)) . (x - h)) by A7, Th4

.= 0 by A5, A6 ;

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

proof

for n being Nat holds S
let x be Real; :: thesis: ((bdif (f,h)) . (0 + 1)) . x = 0

thus ((bdif (f,h)) . (0 + 1)) . x = (bD (((bdif (f,h)) . 0),h)) . x by Def7

.= (bD (f,h)) . x by Def7

.= (f . x) - (f . (x - h)) by Th4

.= 0 by A2 ; :: thesis: verum

end;thus ((bdif (f,h)) . (0 + 1)) . x = (bD (((bdif (f,h)) . 0),h)) . x by Def7

.= (bD (f,h)) . x by Def7

.= (f . x) - (f . (x - h)) by Th4

.= 0 by A2 ; :: thesis: verum

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