let F, G be Field; :: thesis: for V being VectSp of F

for W being VectSp of G

for f being Function of V,W

for h being Element of V

for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let V be VectSp of F; :: thesis: for W being VectSp of G

for f being Function of V,W

for h being Element of V

for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let W be VectSp of G; :: thesis: for f being Function of V,W

for h being Element of V

for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let f be Function of V,W; :: thesis: for h being Element of V

for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let h be Element of V; :: thesis: for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let n be Nat; :: thesis: ( f is constant implies for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W )

assume A1: f is constant ; :: thesis: for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

A2: for x being Element of V holds (f /. (x + h)) - (f /. x) = 0. W

for W being VectSp of G

for f being Function of V,W

for h being Element of V

for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let V be VectSp of F; :: thesis: for W being VectSp of G

for f being Function of V,W

for h being Element of V

for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let W be VectSp of G; :: thesis: for f being Function of V,W

for h being Element of V

for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let f be Function of V,W; :: thesis: for h being Element of V

for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let h be Element of V; :: thesis: for n being Nat st f is constant holds

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

let n be Nat; :: thesis: ( f is constant implies for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W )

assume A1: f is constant ; :: thesis: for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W

A2: for x being Element of V holds (f /. (x + h)) - (f /. x) = 0. W

proof

for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W
let x be Element of V; :: thesis: (f /. (x + h)) - (f /. x) = 0. W

x + h in the carrier of V ;

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

x in the carrier of V ;

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 + h)) - (f /. x) = 0. W by RLVECT_1:15; :: thesis: verum

end;x + h in the carrier of V ;

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

x in the carrier of V ;

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 + h)) - (f /. x) = 0. W by RLVECT_1:15; :: thesis: verum

proof

hence
for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W
; :: thesis: verum
defpred S_{1}[ Nat] means for x being Element of V holds ((fdif (f,h)) . ($1 + 1)) /. x = 0. W;

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 Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W ; :: 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 Element of V holds ((fdif (f,h)) . (k + 1)) /. x = 0. W ; :: thesis: S_{1}[k + 1]

let x be Element of V; :: thesis: ((fdif (f,h)) . ((k + 1) + 1)) /. x = 0. W

A6: ((fdif (f,h)) . (k + 1)) /. (x + h) = 0. W by A5;

reconsider fdk = (fdif (f,h)) . (k + 1) as Function of V,W by Th2;

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

.= (fD (((fdif (f,h)) . (k + 1)),h)) /. x by Def6

.= (fdk /. (x + h)) - (fdk /. x) by Th3

.= (0. W) - (0. W) by A5, A6

.= 0. W by RLVECT_1:15 ;

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

end;assume A5: for x being Element of V holds ((fdif (f,h)) . (k + 1)) /. x = 0. W ; :: thesis: S

let x be Element of V; :: thesis: ((fdif (f,h)) . ((k + 1) + 1)) /. x = 0. W

A6: ((fdif (f,h)) . (k + 1)) /. (x + h) = 0. W by A5;

reconsider fdk = (fdif (f,h)) . (k + 1) as Function of V,W by Th2;

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

.= (fD (((fdif (f,h)) . (k + 1)),h)) /. x by Def6

.= (fdk /. (x + h)) - (fdk /. x) by Th3

.= (0. W) - (0. W) by A5, A6

.= 0. W by RLVECT_1:15 ;

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

proof

for n being Nat holds S
let x be Element of V; :: thesis: ((fdif (f,h)) . (0 + 1)) /. x = 0. W

thus ((fdif (f,h)) . (0 + 1)) /. x = (fD (((fdif (f,h)) . 0),h)) /. x by Def6

.= (fD (f,h)) /. x by Def6

.= (f /. (x + h)) - (f /. x) by Th3

.= 0. W by A2 ; :: thesis: verum

end;thus ((fdif (f,h)) . (0 + 1)) /. x = (fD (((fdif (f,h)) . 0),h)) /. x by Def6

.= (fD (f,h)) /. x by Def6

.= (f /. (x + h)) - (f /. x) by Th3

.= 0. W by A2 ; :: thesis: verum

hence for x being Element of V holds ((fdif (f,h)) . (n + 1)) /. x = 0. W ; :: thesis: verum