Images of not Lindelِf spaces and their squares

If X is not Lindelِf and either X is locally compact or some closed A in X satisfies d(A) < L(A), then X2 has a not realcompact continuous image. If X is locally Lindelِf but not Lindelِf or if X is not linearly Lindelِf then either X2 has a not realcompact continuous image or some closed subspace of X maps continuously onto a regular uncountable cardinal. If X is either locally Lindelِf or not linearly Lindelِf and X is normal, then X2 has a not realcompact continuous image.