Theorem 5.3. Let x and y be real numbers.Then xyl =Ixllyl. 问题3：这种证明方法 为什么被称为分情形 Proof. 证明法？ First,suppose that x 0 and y 0.Then xy 0 and we have Ixyl xy,lxl=x,and lyl y.Therefore, Ixyl xy lxllyl, and we have established the result in this case. Second,suppose that x<0 andy 0.Then xy 0 and we have Ixyl =xy,IxI =-x,and lyl =-y.Therefore, 间题5：有的时候，我 们在证明时会用到 ==(-x)(一0=xl, “不失一性”这个 and we have the result for this case as well. 词，你理解这是什么 Third,suppose that either x =0 or y =0.Then xy =0 and we 意思吗？ have lxyl =0,and either Ixl=0 or lyl =0.Therefore, Ixyl =0=lxllyl. establishing the result in this case too. For our final case,suppose that one number is positive and the 问题4：这种证明方法 other is negative.Thus,we may assume that x 0 and y>0.Then xy 0 and we have lxyl =-(xy),Ixl =-x,and lyl =y.Therefore, 最“令人担心”的是 什么？ Ixy=-(xy)=(xy=Ixllyl ● we have now established the result for all four possible cases and we may conclude that xyl =xllyl for all real numbers x and y.问题3：这种证明方法 为什么被称为分情形 证明法？ 问题4：这种证明方法 最“令人担心”的是 什么？ 问题5：有的时候，我 们在证明时会用到 “不失一般性”这个 词，你理解这是什么 意思吗？