theorem
  G is Abelian addGroup iff the addF of G is commutative
proof
  thus G is Abelian addGroup implies the addF of G is commutative
  proof
    assume
A1: G is Abelian addGroup;
    let a,b;
    thus (the addF of G).(a,b) = a + b .= b + a by A1,Lm1
      .= (the addF of G).(b,a);
  end;
  assume
A2: the addF of G is commutative;
  G is Abelian by A2;
  hence thesis;
end;
