 reserve n,m,i,p for Nat,
         h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
 reserve f,f1,f2,g for Function of REAL,REAL;

theorem
  fdif(f,h).(n+1).x = fdif(f,h).n.(x+h)-fdif(f,h).n.x
proof
  defpred X[Nat] means
  fdif(f,h).($1+1).x = fdif(f,h).$1.(x+h)-fdif(f,h).$1.x;
A1: for i st X[i] holds X[i+1]
    proof
      let i;
      assume
      fdif(f,h).(i+1).x = fdif(f,h).i.(x+h)-fdif(f,h).i.x;
A2:   fdif(f,h).(i+1) is Function of REAL,REAL by DIFF_1:2;
      fdif(f,h).(i+2).x = fdif(f,h).(i+1+1).x
        .= fD(fdif(f,h).(i+1),h).x by DIFF_1:def 6
        .= fdif(f,h).(i+1).(x+h)-fdif(f,h).(i+1).x by A2,DIFF_1:3;
      hence thesis;
    end;
A3:X[0] by Th40;
  for n holds X[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
