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 fD(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;
  fD(f,h).x = f.(x+h)-f.x by DIFF_1:3
    .= a*(x+h)^2+b*(x+h)+c-f.x by A1
    .= a*(x+h)^2+b*(x+h)+c-(a*x^2+b*x+c) by A1
    .= 2*a*h*x+a*h^2+b*h;
  hence thesis;
end;
