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
  (fdif(f,h).n).x = (bdif(f,h).n).(x+n*h)
proof
  defpred X[Nat] means for x holds (fdif(f,h).$1).x = (bdif(f,h).$1).(x+$1*h);
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume
A2: for x holds (fdif(f,h).k).x = (bdif(f,h).k).(x+k*h);
    let x;
A3: (fdif(f,h).k).(x+h) = (bdif(f,h).k).(x+h+k*h) by A2;
A4: fdif(f,h).k is Function of REAL,REAL by Th2;
A5: bdif(f,h).k is Function of REAL,REAL by Th12;
    (fdif(f,h).(k+1)).x = fD(fdif(f,h).k,h).x by Def6
      .= (fdif(f,h).k).(x+h) - (fdif(f,h).k).x by A4,Th3
      .= (bdif(f,h).k).(x+h+k*h) - (bdif(f,h).k).(x+k*h) by A2,A3
      .= (bdif(f,h).k).(x+(k+1)*h) - (bdif(f,h).k).((x+(k+1)*h)-h)
      .= bD(bdif(f,h).k,h).(x+(k+1)*h) by A5,Th4
      .= (bdif(f,h).(k+1)).(x+(k+1)*h) by Def7;
    hence thesis;
  end;
A6: X[0]
  proof
    let x;
    (fdif(f,h).0).x = f.x by Def6
      .= (bdif(f,h).0).(x+0*h) by Def7;
    hence thesis;
  end;
  for n holds X[n] from NAT_1:sch 2(A6,A1);
  hence thesis;
end;
