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 Th6:
  for X being set, SF being Subset-Family of [:X,X:] st X = {{}} &
  SF = {[:X,X:]} holds UniformSpaceStr(# X,SF #) is UniformSpace
  proof
    set X = {{}};
    reconsider SF = {[:X,X:]} as Subset-Family of [:X,X:] by ZFMISC_1:68;
    set US = UniformSpaceStr(# X,SF #);
    now
      for Y1,Y2 be Subset of [:the carrier of US,the carrier of US:] st
         Y1 in the entourages of US & Y1 c= Y2 holds Y2 in the entourages of US
      proof
        let Y1,Y2 be Subset of [:the carrier of US,the carrier of US:];
        assume that
A1:     Y1 in the entourages of US and
A2:     Y1 c= Y2;
A3:     Y1 = [:X,X:] by A1,TARSKI:def 1;
        Y1 = Y2 by A2,A3;
        hence thesis by A1;
      end;
      then the entourages of US is upper;
      hence US is upper;
      for a,b be Subset of [:the carrier of US,the carrier of US:] st
        a in the entourages of US & b in the entourages of US holds
        a /\ b in the entourages of US
      proof
        let a,b be Subset of [:the carrier of US,the carrier of US:];
        assume that
A4:     a in the entourages of US and
A5:     b in the entourages of US;
        a = [:X,X:] & b = [:X,X:] by A4,A5,TARSKI:def 1;
        hence a /\ b in the entourages of US by TARSKI:def 1;
      end;
      hence US is cap-closed by ROUGHS_4:def 2;
      for S being Element of the entourages of US holds id X c= S
      proof
        let S be Element of the entourages of US;
        S = [:X,X:] by TARSKI:def 1;
        hence thesis;
      end;
      hence US is axiom_U1;
      for S being Element of the entourages of US holds
        S[~] in the entourages of US
      proof
        let S be Element of the entourages of US;
        S = [:X,X:] by TARSKI:def 1;
        then S[~] = [:X,X:] by SYSREL:5;
        hence thesis by TARSKI:def 1;
      end;
      hence US is axiom_U2;
      for S being Element of the entourages of US holds
        ex W being Element of the entourages of US st W [*] W c= S
      proof
        let S be Element of the entourages of US;
        take S;
        S = [:X,X:] by TARSKI:def 1;
        hence thesis;
      end;
      hence US is axiom_U3;
    end;
    hence thesis;
  end;
