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 Th36:
  basis_Pervin_uniformity(SF) is cap-closed
  proof
    now
      let x,y be set;
      assume that
A1:   x in basis_Pervin_uniformity(SF) and
A2:   y in basis_Pervin_uniformity(SF);
      consider Y being Subset-Family of [: X,X :] such that
A3:   Y c= subbasis_Pervin_uniformity(SF) and
A4:   Y is finite and
A5:   x = Intersect Y by A1,CANTOR_1:def 3;
      consider Z being Subset-Family of [:X,X:] such that
A6:   Z c= subbasis_Pervin_uniformity(SF) and
A7:   Z is finite and
A8:   y = Intersect Z by A2,CANTOR_1:def 3;
A9:   x /\ y = Intersect (Y \/ Z) by A5,A8,MSSUBFAM:8;
      Y \/ Z c= subbasis_Pervin_uniformity(SF) by A3,A6,XBOOLE_1:8;
      hence x /\ y in basis_Pervin_uniformity(SF) by A9,A4,A7,CANTOR_1:def 3;
    end;
    hence thesis;
  end;
