reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;

theorem Th4:
  for e being Element of TG,
      V being a_neighborhood of 1_TG holds {e} * V is a_neighborhood of e
  proof
    let e be Element of TG,
        V be a_neighborhood of 1_TG;
    consider o be Subset of TG such that
A1: o is open and
A2: o c= V and
A3: 1_TG in o by CONNSP_2:6;
    now
      thus e * o is open by A1;
      thus e * o c= {e} * V
      proof
        let x be object;
        assume
A4:     x in e * o;
        e * o c= e * V by A2,GROUP_3:5;
        hence thesis by A4;
      end;
      e = e * 1_TG by GROUP_1:def 4;
      hence e in e * o by A3,GROUP_2:27;
    end;
    hence thesis by CONNSP_2:6;
  end;
