 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
  cD(cD(f,h),h).x = fD(f,h).x-bD(f,h).x
proof
  cD(cD(f,h),h).x = cD(f,h).(x+h/2)-cD(f,h).(x-h/2) by DIFF_1:5
    .= f.((x+h/2)+h/2)-f.((x+h/2)-h/2)-cD(f,h).(x-h/2) by DIFF_1:5
    .= f.((x+h/2)+h/2)-f.((x+h/2)-h/2)-(f.((x-h/2)+h/2)-f.((x-h/2)-h/2))
                                                             by DIFF_1:5
    .= f.(x+h)-f.x-(f.x-f.(x-h))
    .= fD(f,h).x-(f.x-f.(x-h)) by DIFF_1:3
    .= fD(f,h).x-bD(f,h).x by DIFF_1:4;
  hence thesis;
end;
