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If the lips are gone, the teeth will be cold--  one cannot exist without the other

During the Spring and Autumn Period (chūn qiū 春秋时期,770-476 BC), Duke Xian of the State of Jin Wanted to expand his position of strength and sphere of influence. Therefore he would like to send his troops to destroy the State of Guo on the pretest that the State of Guo often encroached on the borders of the State of Jin. But there was a third state, the State of Yu, between the State of Jin and the State of Guo, and the Jin army had to cross the State of Yu before it could reach the State of Guo. "How can my army cross the State of Yu without a hitch?" Duke Xian of the State of Jin asked his ministers.Xun Xi , one of the ministers, said, "The monarch of the State of Yu is short - sighted and covets small advantages. If we give him priceless precious stones and fine horses, it is not unlikely that he will allow our army to pass through his country. " Seeing that Duke Xian of the State of Jin was a little bit grudging, Xun Xi continued to say, "The State of Yu and the State of Guo are neighbor stated as closely related as lips and teeth. The State of Yu cannot exist independently if the state of Guo is destroyed. Your precious stones and fime hores are just left in the care of the monarch of the State of Yu." So Duke Xian of the State of Jin accepted Xun Xi's plan. When the monarch of Yu saw the precious gifts, he was elated,and readily promised to let the Jin army pass through his state. Hearing the news, Gong Zhiqi, one of the ministers as the State of Yu, hastened to admonish the monarch, saying," That won't do.For the State of Yu and the State of Guo are neighbor states as closely related as lips and teeth. Our two small states are interdependent, and can help each other when problems crop up .If the State of Guo were destroyed, it would be difficult for our State of Yu to continue to exist. As the common saying goes, if the lips are gone ,the teeth will be cold, The teeth can hardly be kept if the lips are gone. So it won't do at all to allow the Jin army pass our state." The monarch of the State of Yu said," The State of Jin is a big state. Now they here specially to present gifts to us with the intention of being on friendly terms with us. Under such circumstances, how can we refuse to allow them to pass through our state?" Hearing this, Gong Zhiqi sighed repeatedly. Knowing that the State of Yu would soon be destroyed, Gong Zhiqi left the State of Yu together with his whole family. As expected, the troops of the State of Jin, allowed to pass through the State of Yu, destroyed the State of Guo and on their return trip captured the monarch of the State of Yu who went out personally to meet them, thus destroying the State of Yu as Well. This story appears in the chapter " The Fifth Year of Duke Xi " in Zuo zhuan,the famous commentary by Zuo Qiuming on The Spring and Autumn Annals. The set phrase " if the lips are gone, the teeth will be cold " is used to mean that two persons or things share a common lot and that is one fails, the other is in danger.