THE BRONX - The City Council says it is moving forward with a lawsuit that aims to halt new restrictions on public homeless shelters that were proposed by the mayor. Mayor Michael Bloomberg's policy would require adults to prove that they have nowhere else to stay before boarding at a city shelter. The new policy, announced last month, has been postponed because of a lawsuit brought against the city by the Legal Aid Society. Bloomberg says his plan would save the city millions, but others argue it would put thousands out in the cold this winter. City Council Speaker Christine Quinn called the policy "cruel and punitive."