
theorem Lemma2To57:
  for a,b being positive Nat, k,x,m being Nat st
    ArProg (b,a).k = x^2 holds
      ArProg (b,a).(m ^2 * a + 2 * m * x + k) is square
  proof
    let a,b be positive Nat, k,x,m be Nat;
    assume
D2: ArProg (b,a).k = x ^2;
D1: ArProg (b,a).k = b + a * k by NUMBER06:7;
    (m * a + x) ^2 = (m * a) ^2 + 2 * m * a * x + x ^2
          .= a * (m ^2 * a + 2 * m * x + k) + b by D1,D2;
    hence thesis by NUMBER06:7;
  end;
