reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;
reserve US for UniformSpace;

theorem Th18:
  for O being open Subset of partition_topology(D) holds O is closed
  proof
    let O be open Subset of partition_topology(D);
    O` in UniCl D by PRE_TOPC:def 2,PROB_1:def 1;
    hence thesis by PRE_TOPC:def 2;
  end;
