theorem
  A <> {} implies (0_G) * A = {0_G}
proof
  set y = the Element of A;
  assume
A1: A <> {};
  then reconsider y as Element of G by TARSKI:def 3;
  thus (0_G) * A c= {0_G}
  proof
    let x be object;
    assume x in (0_G) * A;
    then ex a st x = (0_G) * a & a in A by ThB42;
    then x = 0_G by Th17;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {0_G};
  then x = 0_G by TARSKI:def 1;
  then (0_G) * y = x by Th17;
  hence thesis by A1,ThB42;
end;
