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 Th40:
  for Y being non empty Subset-Family of [:X,X:] st
  Y c= subbasis_Pervin_uniformity(SF) holds Y[~] = Y
  proof
    let Y be non empty Subset-Family of [:X,X:];
    assume
A1: Y c= subbasis_Pervin_uniformity(SF);
A2: Y[~] c= Y
    proof
      let x be object;
      assume x in Y[~];
      then consider y be Element of Y such that
A3:   x = y~;
      y in subbasis_Pervin_uniformity(SF) by A1;
      then consider A be Element of SF such that
A4:   y = block_Pervin_uniformity(A);
      reconsider z = y as Relation of X;
      z~ = y by A4,Th39;
      hence thesis by A3;
    end;
    Y c= Y[~]
    proof
      let x be object;
      assume x in Y;
      then consider y be Element of Y such that
A5:   x = y;
      y in subbasis_Pervin_uniformity(SF) by A1;
      then consider A be Element of SF such that
A6:   y = block_Pervin_uniformity(A);
      reconsider z = y as Relation of X;
      reconsider t = z~ as Element of Y by A6,Th39;
      t~ in Y[~];
      hence thesis by A5;
    end;
    hence thesis by A2;
  end;
