reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;

theorem
  (bdif(bdif(f,h).m,h).n).x = bdif(f,h).(m+n).x
proof
  defpred X[Nat] means
  for x holds (bdif(bdif(f,h).m,h).$1).x = bdif(f,h).(m+$1).x;
A1: X[0] by DIFF_1:def 7;
A2: for i be Nat st X[i] holds X[i+1]
  proof
    let i be Nat;
    assume
A3: for x holds (bdif(bdif(f,h).m,h).i).x = bdif(f,h).(m+i).x;
    let x;
    bdif(f,h).m is Function of REAL,REAL by DIFF_1:12;
    then
A4: bdif(bdif(f,h).m,h).i is Function of REAL,REAL by DIFF_1:12;
A5: bdif(f,h).(m+i) is Function of REAL,REAL by DIFF_1:12;
    bdif(bdif(f,h).m,h).(i+1).x = bD(bdif(bdif(f,h).m,h).i,h).x by DIFF_1:def 7
      .= bdif(bdif(f,h).m,h).i.x - bdif(bdif(f,h).m,h).i.(x-h) by A4,DIFF_1:4
      .= bdif(f,h).(m+i).x - bdif(bdif(f,h).m,h).i.(x-h) by A3
      .= bdif(f,h).(m+i).x - bdif(f,h).(m+i).(x-h) by A3
      .= bD(bdif(f,h).(m+i),h).x by A5,DIFF_1:4
      .= bdif(f,h).(m+i+1).x by DIFF_1:def 7;
    hence thesis;
  end;
  for n be Nat holds X[n] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
