An Awkward Sentence

Hello there,

Can somebody help me with my grammatical analysis on the part "It's an old saying.... is true" in the sentence below, and tell me whether this sentence is acceptable?

"The rich get richer, and the poor get poorer. It's an old saying that a lot of people feel is true"

While the subject of this sentence is "It" which indicates the meaning conveyed in the previous setence, the verb is "is" and "an old saying" is the subject complement for explaining the subject "It".

As for "that a lot of people feel", it is a relative clause for explaining "an old saying". Within this clause, "that" is the relative pronoun, "a lot of people" is the subject, and "feel" is the verb.

Therefore, the corporal verb "is" preceding "true" is redundant, and makes the sentence "It's an old saying...." involving two verbs.

The explaination could be complex, but please tell me how you think about the sentence.

Thanks!!