
theorem Th4:
  for a be Element of COMPLEX holds a*a*' = |.a.|^2+0*<i>
proof
  let z be Element of COMPLEX;
  0<= (Re z)^2 & 0<= (Im z)^2 by XREAL_1:63;
  then (Re z)^2 + (Im z)^2 = |.z.|^2 by SQUARE_1:def 2;
  then
A1: Re (z*z*')=|.z.|^2 by COMPLEX1:40;
  Im (z*z*')=0 by COMPLEX1:40;
  hence thesis by A1,COMPLEX1:13;
end;
