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 Th38:
  basis_Pervin_uniformity(SF) is axiom_UP1
  proof
    for B being Element of basis_Pervin_uniformity(SF) holds id X c= B
    proof
      let B be Element of basis_Pervin_uniformity(SF);
      B in FinMeetCl(subbasis_Pervin_uniformity(SF));
      then consider Y being Subset-Family of [:X,X:] such that
A1:   Y c= subbasis_Pervin_uniformity(SF) and
      Y is finite and
A2:   B = Intersect Y by CANTOR_1:def 3;
      id X c= B
      proof
        let t be object;
        assume
A3:     t in id X;
        then consider a,b be object such that
A4:     t = [a,b] by RELAT_1:def 1;
A5:     a in X & a = b by A3,A4,RELAT_1:def 10;
        per cases;
        suppose Y is empty;
          then B = [:X,X:] by A2,SETFAM_1:def 9;
          hence thesis by A3;
        end;
        suppose
A7:       Y is non empty; then
A8:       Intersect Y = meet Y by SETFAM_1:def 9;
          now
            let y be set;
            assume y in Y;
            then y in the set of all block_Pervin_uniformity(O) where
              O is Element of SF by A1;
            then consider O be Element of SF such that
A9:         y = block_Pervin_uniformity(O);
A10:        [:X \ O,X \ O:] c= y &
              [:O,O:] c= y by A9,XBOOLE_1:10;
            per cases by A5,XBOOLE_0:def 5;
            suppose a in X \ O;
              then [a,b] in [:X \ O,X \ O:]
                by A5,ZFMISC_1:def 2;
              hence t in y by A4,A10;
            end;
            suppose a in O;
              then [a,b] in [:O,O:] by A5,ZFMISC_1:def 2;
              hence t in y by A4,A10;
            end;
          end;
          hence thesis by A2,A8,A7,SETFAM_1:def 1;
        end;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
