reserve X for set,
        A for Subset of X,
        R,S for Relation of X;

theorem Th11:
  for USS being non empty cap-closed axiom_U1 UniformSpaceStr,
  x being Element of USS,
  V,W being Element of the entourages of USS holds
  Neighborhood(V,x) /\ Neighborhood(W,x) = Neighborhood(V /\ W,x)
  proof
    let USS be non empty cap-closed axiom_U1 UniformSpaceStr,
    x be Element of USS,
    V,W be Element of the entourages of USS;
    set NV = Neighborhood(V,x), NW = Neighborhood(W,x),
       NVW = Neighborhood(V /\ W,x);
A1: NV /\ NW c= NVW
    proof
      let t be object;
      assume
A2:   t in NV /\ NW;
      then t in {y where y is Element of USS: [x,y] in V} by XBOOLE_0:def 4;
      then consider y1 be Element of USS such that
A3:   t = y1 and
A4:   [x,y1] in V;
      t in {y where y is Element of USS: [x,y] in W} by A2,XBOOLE_0:def 4;
      then consider y2 be Element of USS such that
A5:   t = y2 and
A6:   [x,y2] in W;
      [x,y1] in V /\ W & [x,y2] in V /\ W by A3,A4,A5,A6,XBOOLE_0:def 4;
      hence thesis by A3;
    end;
    NVW c= NV /\ NW
    proof
      let t be object;
      assume t in NVW;
      then consider y be Element of USS such that
A7:   t = y and
A8:   [x,y] in V /\ W;
A9:   [x,y] in V & [x,y] in W by A8,XBOOLE_0:def 4; then
A10:  t in NV by A7;
      t in {y where y is Element of USS: [x,y] in W} by A7,A9;
      hence t in NV /\ NW by A10,XBOOLE_0:def 4;
    end;
    hence thesis by A1;
  end;
