He is proud that he is rich.

Question

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What is the part of speech of 'that he is rich'? Is it an adverbial clasue? Or is 'of' omitted before 'that'?

Answer 1

"that he is rich" is a noun clause, but for the life of me, I can't remember what this construction is called!

If you want to consider "of" omitted, then this noun cause could be called the object of the understood preposition.

Go here -- http://grammar.ccc.commnet.edu/grammar/clauses.htm#dependent -- and read about noun clauses and the examples given.