reserve X for set,
        A for Subset of X,
        R,S for Relation of X;
reserve QUS for Quasi-UniformSpace;
reserve SUS for Semi-UniformSpace;

theorem Th18:
  for US being non empty upper cap-closed axiom_U1 UniformSpaceStr,
  x being Element of US holds
  Neighborhood(x) is Filter of the carrier of US
  proof
    let US be non empty upper cap-closed axiom_U1 UniformSpaceStr,
    x be Element of US;
    set N = Neighborhood(x);
    now
      thus not {} in N
      proof
        assume {} in N;
        then ex V be Element of the entourages of US st {} = Neighborhood(V,x);
        hence thesis;
      end;
      hereby
        let Y1,Y2 be Subset of US;
        hereby
          assume that
A1:       Y1 in N and
A2:       Y2 in N;
          N is cap-closed by Th17;
          hence Y1 /\ Y2 in N by A1,A2,FINSUB_1:def 2;
        end;
        hereby
          assume that
A3:       Y1 in N and
A4:       Y1 c= Y2;
          N is upper by Th15;
          hence Y2 in N by A3,A4;
        end;
      end;
    end;
    hence thesis by CARD_FIL:def 1;
  end;
