reserve n,m,k,i for Nat,
  h,r,r1,r2,x0,x1,x2,x for Real,
  S for Functional_Sequence of REAL,REAL,
  y for set;
reserve f,f1,f2 for Function of REAL,REAL;

theorem Th12:
  for n be Nat holds bdif(f,h).n is Function of REAL,REAL
proof
  defpred X[Nat] means bdif(f,h).$1 is Function of REAL,REAL;
A1: for k be Nat st X[k] holds X[k+1]
  proof
    let k be Nat;
    assume bdif(f,h).k is Function of REAL,REAL;
    then bD(bdif(f,h).k,h) is Function of REAL,REAL;
    hence thesis by Def7;
  end;
A2: X[0] by Def7;
  for n be Nat holds X[n] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
