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 Th12:
  for TG being non empty commutative TopologicalGroup,
       U being a_neighborhood of 1_TG holds
  element_left_uniformity(U) = element_right_uniformity(U)
  proof
    let TG be non empty commutative TopologicalGroup,
    U be a_neighborhood of 1_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 consider u,v be Element of TG such that
A1:     x = [u,v] and
A2:     u" * v in U;
        v * u" in U by A2,GROUP_1:def  12;
        hence thesis by A1;
      end;
      thus element_right_uniformity(U) c= element_left_uniformity(U)
      proof
        let x be object;
        assume x in element_right_uniformity(U);
        then consider u,v be Element of TG such that
A3:     x = [u,v] and
A4:     v * u" in U;
        u" * v in U by A4,GROUP_1:def  12;
        hence thesis by A3;
      end;
    end;
    hence thesis;
  end;
