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;

theorem
  for X being empty set,cB being empty Subset-Family of [:X,X:] holds
  cB is quasi_basis & cB is axiom_UP1 & cB is axiom_UP2 &
  cB is axiom_UP3
  proof
    let X be empty set,cB be empty Subset-Family of [:X,X:];
    for B1,B2 be Element of cB ex B be Element of cB st B c= B1 /\ B2
    proof
      let B1,B2 be Element of cB;
      reconsider B = {} as Element of cB by SUBSET_1:def 1;
      take B;
      thus thesis;
    end;
    hence cB is quasi_basis;
    for B be Element of cB holds id X c= B;
    hence cB is axiom_UP1;
    for B1 be Element of cB holds ex B2 be Element of cB st B2 c= B1~
    proof
      let B1 be Element of cB;
      reconsider B2 = {} as Element of cB by SUBSET_1:def 1;
      B2 c= B1~;
      hence thesis;
    end;
    hence cB is axiom_UP2;
    for B1 be Element of cB ex B2 be Element of cB st B2 * B2 c= B1
    proof
      let B1 be Element of cB;
      reconsider B2 = {} as Element of cB by SUBSET_1:def 1;
      B2 * B2 c= B1;
      hence thesis;
    end;
    hence cB is axiom_UP3;
  end;
