One proof is that the Holy Spirit is referred to in Scripture as both the Spirit of the Father (Mt 10:20, Rom 8:10-11, 2 Cor 1:21-22, Eph 3:14-16) and as the Spirit of the Son (Rom 8:9, Gal 4:6, Phil 1:19, 1 Pt 1:11). Statements saying that the Spirit is "of" the other two Persons of the Trinity indicate that his Person is tightly bound up with and originates from them (just as the Son is the Son "of" the Father).
A second proof is that the external relations of the Trinity model their internal ones. In John 14:26 the Spirit is said to proceed from the Father, but a chapter later, in 15:26, Jesus states that he will send the Spirit from the Father. The same relation is reflected in Acts 2:33, where Peter states that Jesus has received the Spirit from the Father and sends him.
A philosophical explanation of this is found in the Council of Florence, which stated in 1439, "Since the Father has through generation given to the only-begotten Son everything that belongs to the Father, except being Father, the Son has also eternally from the Father, from whom he is eternally born, that the Holy Spirit proceeds from the Son" (Decree for the Greeks).
The Spirit proceeds from the Father and the Son because the Father has given all things to the Son, including the procession of the Holy Spirit. For more information see the Catechism of the Catholic Church 246-248, 264.