theorem Th15:
  for K being comRing, a1,a2 being Element of K holds
  (a1 + a2)*(a1 - a2) = a1 |^2 - a2 |^2
  proof
    let K be comRing, a1,a2 be Element of K;
    thus (a1 + a2)*(a1 - a2) = a1*(a1-a2)+a2*(a1-a2) by VECTSP_1:def 7
    .= a1*a1-a1*a2+a2*(a1-a2) by VECTSP_1:11
    .= a1 |^2 -a1*a2+a2*(a1-a2) by GROUP_1:51
    .= a1 |^2 -a1*a2+(a2*a1-a2*a2) by VECTSP_1:11
    .= a1 |^2 -a1*a2+(a1*a2-a2 |^2) by GROUP_1:51
    .= a1 |^2 + (-a1*a2 +(a1*a2-a2 |^2)) by ALGSTR_1:7
    .= a1 |^2 + ((-a1*a2+a1*a2)-a2 |^2) by ALGSTR_1:7
    .= a1 |^2 + (0.K-a2 |^2) by VECTSP_1:19
    .= a1 |^2 - a2 |^2 by VECTSP_1:18;
  end;
