How did China's turn to communism would affect the Cold War?

The Cold War was characterised by the tension between the two contending superpowers the United States and the Soviet Union. Yet the position of Communist China under Mao Zedong in the Cold War, in many respects, was not minor but of great importance especially with regards to its relationship with the Soviet Union.

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The rule of law is important for democratic forms of government because if the law is the highest rule, then it applies people without any exception

This would  makes all citizens become equals in the eyes of law, which is  considered to be ideal/highest form of democracy. By putting the rule of law,  All people, regardless of ethnicity, economic status or social backgrounds shall be subjected to the same standard for all of our actions.