theorem
  a1|^2+a1*b1+b1|^2 = (3*(a1+b1)|^2 +(a1-b1)|^2)/4
  proof
    A1: (a1+b1)|^2 = (a1+b1)*(a1+b1) by NEWTON:81
    .= a1*a1 + 2*a1*b1 + b1*b1
    .= a1|^2 + 2*a1*b1 + b1*b1 by NEWTON:81
    .= a1|^2 + 2*a1*b1 + b1|^2 by NEWTON:81;
    (a1-b1)|^2 = (a1-b1)*(a1-b1) by NEWTON:81
    .= a1*a1 - 2*a1*b1 + b1*b1
    .= a1|^2 - 2*a1*b1 + b1*b1 by NEWTON:81
    .= a1|^2 - 2*a1*b1 + b1|^2 by NEWTON:81;
    hence thesis by A1;
  end;
