>>Is the Holy Spirit not part of the Trinity?<<<
Ok. God question. Yes. The Holy Spirit is one of the persons of the trinity.
Is a person of the trinity God? (Yes)
Is a person of the trinity only a part of God?
The Trinity is also professed By Catholics to be One God in three distinct persons which are of the same essence. Each person is God. But neither is it the case that one person in the trinity is not without distinct relations which are incommunicable to the other persons.
The Son is not the Holy Spirit. The Spirit is not the Father, but they are one God.