reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem
  (for x holds f.x = a*x^2+b*x+c) implies for x holds bD(f,h).x = 2*a*h*
  x-a*h^2+b*h
proof
  assume
A1: for x holds f.x = a*x^2+b*x+c;
  let x;
  bD(f,h).x = f.x-f.(x-h) by DIFF_1:4
    .= a*x^2+b*x+c-f.(x-h) by A1
    .= a*x^2+b*x+c-(a*(x-h)^2+b*(x-h)+c) by A1
    .= 2*a*h*x-a*h^2+b*h;
  hence thesis;
end;
