 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
  fD(fD(f,h),h).x = fD(f,2*h).x-2*fD(f,h).x
proof
  fD(fD(f,h),h).x = fD(f,h).(x+h)-fD(f,h).x by DIFF_1:3
    .= (f.((x+h)+h)-f.(x+h))-fD(f,h).x by DIFF_1:3
    .= (f.((x+h)+h)-f.(x+h))-(f.(x+h)-f.x) by DIFF_1:3
    .= (f.(x+2*h)-f.x)-(2*f.(x+h)-2*f.x)
    .= fD(f,2*h).x-2*(f.(x+h)-f.x) by DIFF_1:3
    .= fD(f,2*h).x-2*fD(f,h).x by DIFF_1:3;
  hence thesis;
end;
