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
  G is commutative Group iff for A,g holds A |^ g = A
proof
  thus G is commutative Group implies for A,g holds A |^ g = A by Th55;
  assume
A1: for A,g holds A |^ g = A;
  now
    let a,b;
    {a} = {a} |^ b by A1
      .= {a |^ b} by Th37;
    hence a = a |^ b by ZFMISC_1:3;
  end;
  hence thesis by Th30;
end;
