reserve a, b, c, d, x, y, z for Complex;

theorem Th29:
  z.|.z = |.z.|^2 & |.z.|^2 = Re (z.|.z)
proof
A1: (Re z)^2>=0 & (Im z)^2>=0 by XREAL_1:63;
A2: |.z.|=sqrt((Re z)^2 + (Im z)^2) by COMPLEX1:def 12;
  thus
A3: z.|.z = (Re z)^2+(Im z)^2 by Th28
    .= |.z.|^2 by A1,A2,SQUARE_1:def 2;
  |.z.|^2 = |.z.|^2+(0*<i> qua Complex);
  hence thesis by A3,COMPLEX1:12;
end;
