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 Th39:
  for A being Element of SF, R being Relation of X st
  R = block_Pervin_uniformity(A) holds R~ = block_Pervin_uniformity(A)
  proof
    let A be Element of SF, R be Relation of X;
    assume
    A1: R = block_Pervin_uniformity(A);
    per cases;
    suppose SF is empty; then
      F1: A = {} by SUBSET_1:def 1;
      then R = [:X,X:] by A1,Th34;
      then R~ = [:X,X:] by SYSREL:5;
      hence thesis by F1,Th34;
    end;
    suppose
      SF is non empty;
      then A in SF;
      then reconsider A as Subset of X;
      R~ = [:A,A:] \/ [: X \ A, X \ A:] by A1,Th33;
      hence thesis;
    end;
  end;
