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

theorem
  for USS being non empty axiom_U1 UniformSpaceStr, x being Element of
  the carrier of USS, V being Element of the entourages of USS holds
  x in Neighborhood(V,x)
  proof
    let USS be non empty axiom_U1 UniformSpaceStr,
        x be Element of USS,
        V be Element of the entourages of USS;
    USS is axiom_U1; then
A1: id the carrier of USS c= V;
    [x,x] in id the carrier of USS by RELAT_1:def 10;
    hence thesis by A1;
  end;
