reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

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;
