theorem
  (1).G is commutative
proof
  let a,b be Element of (1).G;
  a in the carrier of (1).G;
  then a in {1_G} by GROUP_2:def 7;
  then
A1: a = 1_G by TARSKI:def 1;
  b in the carrier of (1).G;
  then b in {1_G} by GROUP_2:def 7;
  hence thesis by A1,TARSKI:def 1;
end;
