
theorem
  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) = (m * a + x) ^2
  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 = a * k + b by NUMBER06:7;
    set mm = m;
    (mm * a + x) ^2 = (mm * a) ^2 + 2 * mm * a * x + x ^2
          .= a * (mm ^2 * a + 2 * mm * x + k) + b by D1,D2;
    hence thesis by NUMBER06:7;
  end;
