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

theorem
  for SUS being non empty axiom_U1 UniformSpaceStr, x being Element of
  the carrier of SUS, V being Element of the entourages of SUS holds
  Neighborhood(V,x) = V.:{x} & Neighborhood(V,x) = rng(V|{x}) &
  Neighborhood(V,x) = Im(V,x) & Neighborhood(V,x) = Class(V,x) &
  Neighborhood(V,x) = neighbourhood(x,V)
  proof
    let SUS be non empty axiom_U1 UniformSpaceStr, x be Element of
      the carrier of SUS, V be Element of the entourages of SUS;
    thus Neighborhood(V,x) = V.:{x}
    proof
      thus Neighborhood(V,x) c= V.:{x}
      proof
        let t be object;
        assume t in Neighborhood(V,x);
        then consider y be Element of SUS such that
A3:     t = y and
A4:     [x,y] in V;
        x in {x} by TARSKI:def 1;
        hence thesis by A3,A4,RELAT_1:def 13;
      end;
      let t be object;
      assume t in V.:{x};
      then consider v be object such that
A5:   [v,t] in V and
A6:   v in {x} by RELAT_1:def 13;
A7:   t in the carrier of SUS by A5,ZFMISC_1:87;
      [x,t] in V by A5,A6,TARSKI:def 1;
      hence thesis by A7;
    end;
    hence thesis by RELAT_1:115;
  end;
