First sentence: "A time of prosperity and peace"-- This is not a sentence; it is a noun phrase 'A time' is a noun phrase, where 'a' is the determiner and 'time' is the head. 'Of' is a preposition, so I think it's a prepositional phrase with a noun complement. -- OK, but notice that the prepositional phrase is post-modifying 'time', making the whole thing a larger noun phrase.
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