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 holds (fdif (f,h)) . n is Function of V,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 holds (fdif (f,h)) . n is Function of V,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 holds (fdif (f,h)) . n is Function of V,W

let f be Function of V,W; :: thesis: for h being Element of V
for n being Nat holds (fdif (f,h)) . n is Function of V,W

let h be Element of V; :: thesis: for n being Nat holds (fdif (f,h)) . n is Function of V,W
defpred S1[ Nat] means (fdif (f,h)) . \$1 is Function of V,W;
A1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume (fdif (f,h)) . k is Function of V,W ; :: thesis: S1[k + 1]
then fD (((fdif (f,h)) . k),h) is Function of V,W ;
hence S1[k + 1] by Def6; :: thesis: verum
end;
A2: S1[ 0 ] by Def6;
for n being Nat holds S1[n] from NAT_1:sch 2(A2, A1);
hence for n being Nat holds (fdif (f,h)) . n is Function of V,W ; :: thesis: verum