In the previous papers for the integral Fredgholm equation of the first kind with a degenerate kernel the generalized Heart Bracelet Gift Box (non regularizing) version of the projection method was developed, based on the arbitrary choice of the basis combined with some a priori estimations for the error of the solution.These estimations depend on the residual Spa Chlorine Generators which was neglected earlier.The results of the present paper admit to take the residual into account and show that, generally speaking, this accounting is essential.