theorem Th1:
  x in (1).G iff x = 1_G
proof
  thus x in (1).G implies x = 1_G
  proof
    assume x in (1).G;
    then x in the carrier of (1).G by STRUCT_0:def 5;
    then x in {1_G} by GROUP_2:def 7;
    hence thesis by TARSKI:def 1;
  end;
  thus thesis by GROUP_2:46;
end;
