reserve a,b,c,d for Real;
reserve z,z1,z2 for Complex;

theorem
  Re(z*z*') = (Re z)^2 + (Im z)^2 & Im(z*z*') = 0
proof
  thus Re(z*(z*')) = Re z * Re (z*') - Im z * Im (z*') by Th9
    .= Re z * Re z - Im z * Im (z*') by Th27
    .= Re z * Re z - Im z * -Im z by Th27
    .= (Re z)^2 + (Im z)^2;
  thus Im(z*(z*')) = Re z * Im (z*') + Re (z*') * Im z by Th9
    .= Re z * -Im z + Re (z*') * Im z by Th27
    .= -Re z * Im z + Re z * Im z by Th27
    .= 0;
end;
