That's the best explanation I can provide, Free.
How can that be the best explanation? Equating the Holy Spirit with a mathematical operation doesn't really explain anything; it just makes the Holy Spirit an abstract concept. What does explain things is Scripture, such as the following passages:
Act 16:6 And they went through the region of Phrygia and Galatia, having been forbidden by
the Holy Spirit to speak the word in Asia.
Act 16:7 And when they had come up to Mysia, they attempted to go into Bithynia, but
the Spirit of Jesus did not allow them. (ESV)
Rom 8:9 You, however, are not in the flesh but in the Spirit, if in fact
the Spirit of God dwells in you. Anyone who does not have
the Spirit of Christ does not belong to him. (ESV)
Gal 4:6 And because you are sons, God has sent
the Spirit of his Son into our hearts, crying, "Abba! Father!" (ESV)
Php 1:19 for I know that through your prayers and the help of
the Spirit of Jesus Christ this will turn out for my deliverance, (ESV)
Heb 10:15 And
the Holy Spirit also bears witness to us;
for after saying,
Heb 10:16 “This is the covenant that I will make with them after those days, declares
the Lord: I will put my laws on their hearts, and write them on their minds,”
Heb 10:17
then he adds, “I will remember their sins and their lawless deeds no more.” (ESV)
It's worth noting that Hebrews equates the Holy Spirit with Yahweh, speaking as Yahweh in the OT.
1Pe 1:10 Concerning this salvation, the prophets who prophesied about the grace that was to be yours searched and inquired carefully,
1Pe 1:11 inquiring what person or time
the Spirit of Christ in them was indicating when he predicted the sufferings of Christ and the subsequent glories. (ESV)
It's also worth noting here that we similarly see that the OT prophets prophesied by the "Spirit of Christ." So, the Holy Spirit is known also as the Spirit of God, the Spirit of Christ, the Spirit of Jesus Christ, and the Spirit of his Son, among other names. And, it was by the Holy Spirit that the OT prophets prophesied.
To keep things short, the Holy Spirit:
Acts: Matt 4:1; Acts 8:39, 16:7
Listens: John 16:13
Speaks: John 16:13-15; Acts 1:16, 10:19, 11:12, 11:28, 13:2, 15:28; 1 Tim 4:1; Heb 3:7
Can be lied to: Acts 5:3, which is the same as lying to God (5:9)
Bears witness: Rom 8:16; Heb 10:15; 1 John 5:6
Helps, intercedes, and searches: John 14:16, 15:26, 16:7; Rom 8:26-27; 1 Cor 2:10
Teaches: Luke 12:12; John 16:13; 1 Cor 2:13
Gives gifts: Acts 20:28; 1 Cor 12:11; Heb 2:4
Leads: John 16:13; Gal 5:18, Heb 9:8
Can be grieved: Eph 4:30
Can be outraged: Heb 10:29
Can be blasphemed: Matt 12:31-32
Convicts: John 16:8-11
These are all actions of personal agency. And on it goes. There is still much that can be said, but importantly:
Joh 14:16 And I will ask the Father, and he will give you
another Helper, to be with you forever,
Joh 14:17 even the Spirit of truth, whom the world cannot receive, because it neither sees him nor knows him. You know him, for
he dwells with you and will be in you.
...
Joh 14:23 Jesus answered him, “If anyone loves me, he will keep my word, and my Father will love him, and
we will come to him and make our home with him. (ESV)
Apart from the significance of
paraklētos, what it means and where else it is used in the NT, we see all three persons in the believer. The best, most likely conclusion based on a plain reading of Scripture, is that the Holy Spirit is also truly God but is neither the Father nor the Son but equal to them.