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 cD(f,h).x = 2*a*h* x+b*h
proof
  assume
A1: for x holds f.x = a*x^2+b*x+c;
  let x;
  cD(f,h).x = f.(x+h/2) - f.(x-h/2) by DIFF_1:5
    .= a*(x+h/2)^2+b*(x+h/2)+c - f.(x-h/2) by A1
    .= a*(x+h/2)^2+b*(x+h/2)+c -(a*(x-h/2)^2+b*(x-h/2)+c) by A1
    .= 2*a*h*x+b*h;
  hence thesis;
end;
