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
  for TG being non empty TopologicaladdGroup st TG is Abelian holds
  left_uniformity(TG) = right_uniformity(TG)
  proof
    let TG be non empty TopologicaladdGroup;
    assume
A1: TG is Abelian;
    set X = the set of all element_left_uniformity(U) where
      U is a_neighborhood of 0_TG;
    set Y = the set of all element_right_uniformity(U) where
      U is a_neighborhood of 0_TG;
    now
      thus X c= Y
      proof
        let x be object;
        assume x in X;
        then consider U be a_neighborhood of 0_TG such that
A2:     x = element_left_uniformity(U);
        x = element_right_uniformity(U) by A1,A2,Th13;
        hence thesis;
      end;
      thus Y c= X
      proof
        let x be object;
        assume x in Y;
        then consider U be a_neighborhood of 0_TG such that
A3:     x = element_right_uniformity(U);
        x = element_left_uniformity(U) by A1,A3,Th13;
        hence thesis;
      end;
    end;
    then system_left_uniformity(TG) = system_right_uniformity(TG);
    hence thesis;
  end;
