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 Th13:
  for TG being Abelian TopaddGroup, U being a_neighborhood of 0_TG holds
  element_left_uniformity(U) = element_right_uniformity(U)
  proof
    let TG be Abelian TopaddGroup,
    U be a_neighborhood of 0_TG;
    now
      thus element_left_uniformity(U) c= element_right_uniformity(U)
      proof
        let x be object;
        assume x in element_left_uniformity(U);
        then ex u,v be Element of TG st x = [u,v] & (-u) + v in U;
        hence thesis;
      end;
      thus element_right_uniformity(U) c= element_left_uniformity(U)
      proof
        let x be object;
        assume x in element_right_uniformity(U);
        then ex u,v be Element of TG st x = [u,v] & v + (-u) in U;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
