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 Th17:
  for US being non empty cap-closed axiom_U1 UniformSpaceStr,
  x being Element of US holds Neighborhood(x) is cap-closed
  proof
    let US be non empty cap-closed axiom_U1 UniformSpaceStr,
    x be Element of US;
    set N = Neighborhood(x);
    now
      let Y1,Y2 be set;
      assume that
A1:   Y1 in N and
A2:   Y2 in N;
      consider V1 be Element of the entourages of US such that
A3:   Y1 = Neighborhood(V1,x) by A1;
      consider V2 be Element of the entourages of US such that
A4:   Y2 = Neighborhood(V2,x) by A2;
      Y1 /\ Y2 = Neighborhood(V1 /\ V2,x) by A3,A4,Th11;
      hence Y1 /\ Y2 in N;
    end;
    hence thesis by FINSUB_1:def 2;
  end;
