let h be Real; :: thesis: for f being Function of REAL,REAL
for n being Nat holds (bdif (f,h)) . n is Function of REAL,REAL
let f be Function of REAL,REAL; :: thesis: for n being Nat holds (bdif (f,h)) . n is Function of REAL,REAL
defpred S₁[ Nat] means (bdif (f,h)) . $1 is Function of REAL,REAL;
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 (bdif (f,h)) . k is Function of REAL,REAL ; :: thesis: S₁[k + 1]
then bD (((bdif (f,h)) . k),h) is Function of REAL,REAL ;
hence S₁[k + 1] by Def7; :: thesis: verum

end;

A2: S₁[ 0 ] by Def7;
for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A1);
hence for n being Nat holds (bdif (f,h)) . n is Function of REAL,REAL ; :: thesis: verum