 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
  bD(bD(f,h),h).x = 2*bD(f,h).x-bD(f,2*h).x
proof
  bD(bD(f,h),h).x = bD(f,h).x-bD(f,h).(x-h) by DIFF_1:4
    .= f.x-f.(x-h)-bD(f,h).(x-h) by DIFF_1:4
    .= f.x-f.(x-h)-(f.(x-h)-f.((x-h)-h)) by DIFF_1:4
    .= 2*(f.x-f.(x-h))-(f.x-f.(x-2*h))
    .= 2*bD(f,h).x-(f.x-f.(x-2*h)) by DIFF_1:4
    .= 2*bD(f,h).x-bD(f,2*h).x by DIFF_1:4;
  hence thesis;
end;
