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;
reserve R for Relation of X;
reserve SF for Subset-Family of X, A for Element of SF;

theorem
  for R being Relation of X st
  R is Element of subbasis_Pervin_uniformity(SF) holds
  R~ is Element of subbasis_Pervin_uniformity(SF)
  proof
    let R be Relation of X;
    assume
A1: R is Element of subbasis_Pervin_uniformity(SF);
    then R in the set of all block_Pervin_uniformity(A) where
      A is Element of SF;
    then ex A be Element of SF st R = block_Pervin_uniformity(A);
    hence thesis by A1,Th39;
  end;
