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 Th15:
  for USS being non empty upper axiom_U1 UniformSpaceStr, x being Element of
    the carrier of USS holds Neighborhood(x) is upper
  proof
    let US be non empty upper axiom_U1 UniformSpaceStr,
        x be Element of US;
    set N = Neighborhood(x);
    now
      let S1,S2 be Subset of US;
      assume that
A1:   S1 in N and
A2:   S1 c= S2;
      consider V1 be Element of the entourages of US such that
A3:   S1 = Neighborhood(V1,x) by A1;
A4:   V1 c= [:{x},S1:] \/ ([:the carrier of US,the carrier of US:] \
        [:{x},the carrier of US:])
      proof
        let t be object;
        assume
A5:     t in V1;
        consider a,b be object such that
        a in the carrier of US and
A7:     b in the carrier of US and
A8:     t = [a,b] by A5,ZFMISC_1:def 2;
        reconsider b as Element of US by A7;
        per cases;
        suppose
A9:       x = a;
          then x in {x} & b in S1 by A3,A5,A8,TARSKI:def 1; then
A10:      t in [:{x},S1:] by A9,A8,ZFMISC_1:def 2;
          [:{x},S1:] c= [:{x},S1:] \/
          ([:the carrier of US,the carrier of US:] \ [:{x},the carrier of US:])
            by XBOOLE_1:7;
          hence thesis by A10;
        end;
        suppose
A11:      not x = a;
A12:      not [a,b] in [:{x},the carrier of US:]
          proof
            assume [a,b] in [:{x},the carrier of US:];
            then ex c,d be object st c in {x} & d in the carrier of US &
             [a,b] = [c,d] by ZFMISC_1:def 2;
            then a in {x} & b in the carrier of US by XTUPLE_0:1;
            hence thesis by A11,TARSKI:def 1;
          end;
A13:      t in ([:the carrier of US,the carrier of US:] \
            [:{x},the carrier of US:]) by A5,A12,A8,XBOOLE_0:def 5;
          [:the carrier of US,the carrier of US:] \ [:{x},the carrier of US:]
            c= [:{x},S1:] \/ ([:the carrier of US,the carrier of US:] \
               [:{x},the carrier of US:]) by XBOOLE_1:7;
          hence thesis by A13;
        end;
      end;
      reconsider V2 = [:{x},S2:] \/ ([:the carrier of US,the carrier of US:]
        \ [:{x},the carrier of US:]) as Subset of
        [:the carrier of US,the carrier of US:];
A15:  V1 c= V2
      proof
        [:{x},S1:] c= [:{x},S2:] by A2,ZFMISC_1:95;
        then [:{x},S1:] \/ ([:the carrier of US,the carrier of US:] \
          [:{x},the carrier of US:]) c= [:{x},S2:] \/
          ([:the carrier of US,the carrier of US:] \
          [:{x},the carrier of US:]) by XBOOLE_1:9;
        hence thesis by A4;
      end;
      US is upper;
      then the entourages of US is upper;
      then reconsider V2 as Element of the entourages of US by A15;
      S2 = Neighborhood(V2,x)
      proof
A16:   S2 c= Neighborhood(V2,x)
       proof
         let t be object;
         assume
A17:     t in S2;
         then reconsider t1 = t as Element of US;
A18:     x in {x} by TARSKI:def 1;
         [x,t1] in [:{x},S2:] by A17,A18,ZFMISC_1:def 2;
         then [x,t1] in V2 by XBOOLE_0:def 3;
         hence thesis;
       end;
       Neighborhood(V2,x) c= S2
       proof
         let t be object;
         assume t in Neighborhood(V2,x);
         then ex y0 be Element of US st t = y0 & [x,y0] in V2;
         then per cases by XBOOLE_0:def 3;
         suppose [x,t] in [:{x},S2:];
           then ex a,b be object st a in {x} & b in S2 & [x,t] = [a,b]
              by ZFMISC_1:def 2;
           hence thesis by XTUPLE_0:1;
         end;
         suppose
A19:       [x,t] in [:the carrier of US,the carrier of US:] \
             [:{x},the carrier of US:];
           then ex a,b be object st a in the carrier of US &
             b in the carrier of US & [x,t] = [a,b] by ZFMISC_1:def 2; then
A20:       t in the carrier of US by XTUPLE_0:1;
           x in {x} by TARSKI:def 1;
           then [x,t] in [:{x},the carrier of US:] by A20,ZFMISC_1:def 2;
           hence thesis by A19,XBOOLE_0:def 5;
         end;
       end;
       hence thesis by A16;
     end;
     hence S2 in N;
   end;
   hence thesis;
  end;
