7 - Jesus is like Melchizedek: king of righteousness and peace without earthly genealogy. Jesus is greater than Levi and Abraham, who paid tithes to Melchizedek. He is better than Aaron's priests, since He lives forever, is sinless Himseelf, and was made priest by God's direct oath.
8 - Jesus offers a better sacrifice in the true temple, as mediuator of a better covenant with bettter promises. Looking for a better covenant is justified in the old covenant by Jeremiah 31:31-34, quoted in full.
9 - Jesus entered the holy of holies by His own blood, so the present temple system of sacrifices in the holy place is not needed anymore. Jesus offers a better, more effective, death than the temple sacrifices. Done only once, it removes sins forever, unlike the temple sacrifices.
10 - Jesus did God's will and sat down after serving. Priests sin, and continually offer sacrifice, always standing before God as servants. So let us draw near to Christ, stop sinning, and endure in faith.
11 - The saints of old lived and acted by faith.
12 - Look to Jesus who also suffered, and endure it as God's discipline of His children for our good. Maintain your purity, for you have come to Zion which obligations (greater punishment for rejecting the greater Jesus) and privileges (receiving a greater kingdom) are better and greater than Sinai.
13 - Offer pure sacrifices to God: show hospitality, visit prisoners, stay pure, content and true, bear Christ's reproach, obey your leaders, pray for us.