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 ((bdif (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 ((bdif (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 ((bdif (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 ((bdif (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 ((bdif (f,h)) . (n + 1)) /. x = 0. W
let n be Nat; :: thesis: ( f is constant implies for x being Element of V holds ((bdif (f,h)) . (n + 1)) /. x = 0. W )
assume A1: f is constant ; :: thesis: for x being Element of V holds ((bdif (f,h)) . (n + 1)) /. x = 0. W
A2: for x being Element of V holds (f /. x) - (f /. (x - h)) = 0. W

proof

let x be Element of V; :: thesis: (f /. x) - (f /. (x - h)) = 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) - (f /. (x - h)) = 0. W by RLVECT_1:15; :: thesis: verum

end;

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

proof

defpred S₁[ Nat] means for x being Element of V holds ((bdif (f,h)) . ($1 + 1)) /. x = 0. W;
A4: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: for x being Element of V holds ((bdif (f,h)) . (k + 1)) /. x = 0. W ; :: thesis: S₁[k + 1]
let x be Element of V; :: thesis: ((bdif (f,h)) . ((k + 1) + 1)) /. x = 0. W
A6: ((bdif (f,h)) . (k + 1)) /. (x - h) = 0. W by A5;
A7: (bdif (f,h)) . (k + 1) is Function of V,W 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. W) - (0. W) by A5, A6
.= 0. W by RLVECT_1:15 ;
hence ((bdif (f,h)) . ((k + 1) + 1)) /. x = 0. W ; :: thesis: verum

end;

A8: S₁[ 0 ]

proof

let x be Element of V; :: thesis: ((bdif (f,h)) . (0 + 1)) /. x = 0. W
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. W by A2 ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A8, A4);
hence for x being Element of V holds ((bdif (f,h)) . (n + 1)) /. x = 0. W ; :: thesis: verum

end;

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