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 Th43:
  basis_Pervin_uniformity(SF) is axiom_UP2
  proof
    let B1 be Element of basis_Pervin_uniformity(SF);
    B1 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: B1 = Intersect Y by CANTOR_1:def 3;
    per cases;
    suppose Y is empty; then
A3:   B1 = [:X,X:] by A2,SETFAM_1:def 9;
      B1 c= B1~
      proof
        let x be object;
        assume x in B1;
        then consider a,b be object such that
A4:     a in X and
A5:     b in X and
A6:     x = [a,b] by A2,ZFMISC_1:def 2;
        [b,a] in B1 by A3,A4,A5,ZFMISC_1:def 2;
        hence thesis by A6,RELAT_1:def 7;
      end;
      hence thesis;
    end;
    suppose
A9:   Y is non empty; then
A10:  B1 = meet Y by A2,SETFAM_1:def 9;
      set Y2 = Y[~];
      Y[~] = Y by A1,A9,Th40;
      then reconsider B2 = meet Y2 as Element of basis_Pervin_uniformity(SF)
        by A9,A2,SETFAM_1:def 9;
      B2 c= B1~
      proof
        let x be object;
        assume x in B2;
        then x in meet Y by A1,A9,Th40;
        hence thesis by A10,A1,A9,Th42;
      end;
      hence thesis;
    end;
  end;
