
theorem Th47:
  for q be G_RAT holds Norm(q) = |. Re q .|^2 + |. Im q .| ^2
  proof
    let q be G_RAT;
    A1: |. Re q .|^2 = (Re q)^2 by COMPLEX1:75;
    thus
    Norm(q)=(Re q+Im q*<i>)*(Re q-(Im q)*<i>) by COMPLEX1:13
    .= (Re q)^2 + (Im q)^2
    .= |. Re q .|^2 + |. Im q .| ^2 by A1,COMPLEX1:75;
  end;
