AND THE HOLY SPIRIT ALSO BEARS WITNESS TO US FOR AFTER SAYING: Marturei (3SPAI) de hemin kai to pneuma to hagion meta gar to eirekenai (RAN): (2:3,4; 3:7; 9:8; 2 Samuel 23:2; Nehemiah 9:30; John 15:26; Acts 28:25; 1 Peter 1:11,12; 2 Peter 1:21; Revelation 2:7,11,17,29; 3:6,13,22; 19:10)

This verse shows that the Holy Spirit is our Teacher, the One Jesus promised declaring that...

When the Helper comes, Whom I will send to you from the Father, that is the Spirit of truth, who proceeds from the Father, He will bear witness of Me (Jn 15:26)

The Spirit reveals God the Father and God the Son by inspiring the writing of Scripture. The Spirit has revealed in the past

The Holy Spirit is signifying this, that the way into the holy place has not yet been disclosed while the outer tabernacle is still standing (see note Hebrews 9:8)

He reveals in the present as in this verse (bears witness = present tense = ongoing activity)...

Therefore, just as the Holy Spirit says, "TODAY IF YOU HEAR HIS VOICE (see note Hebrews 3:7)

The Spirit who first led people to write the Scriptures Peter recording that

no prophecy was ever made by an act of human will, but men moved by the Holy Spirit spoke from God (see note 2 Peter 1:21)

The Spirit continues to guide God's people to understand them, as Hebrews shows. The Spirit is the Inspirer and Teacher of Scriptures and the One Who enables us to live according to the Scriptures.

Bears witness (3140) (martureo from mártus = witness, one who has information or knowledge of something & hence can bring to light or confirm something; English ~ martyr) ) means to be a witness, to testify, to give evidence, to give testimony, to affirm that one has seen or heard or experienced something. To be well reported. It means to provide information about a person or an event concerning which the speaker has direct knowledge. Martureo in some context is used in the sense of making an important and solemn declaration. It can be used in the sense of confirmation or approval and so to affirm n a supportive manner.

Martureo is another key word in Hebrews, with 7/81 (almost 10% of the NT uses)

Hebrews 7:8 (note) - In this case mortal men receive tithes, but in that case one receives them, of whom it is witnessed that he lives on.

Hebrews 7:17 (note) - For it is attested of Him, "YOU AREA PRIEST FOREVER ACCORDING TO THE ORDER OF MELCHIZEDEK ."

Hebrews 10:15 (note) - And the Holy Spirit also testifies to us; for after saying,

Hebrews 11:2 (note) - For by it the men of old gained approval.

Hebrews 11:4 (note) - By faith Abel offered to God a better sacrifice than Cain, through which he obtained the testimony that he was righteous, God testifying about his gifts, and through faith, though he is dead, he still speaks.

Hebrews 11:5 (note) - By faith Enoch was taken up so that he would not see death; AND HE WAS NOT FOUND BECAUSE GOD TOOK HIM UP; for he obtained the witness that before his being taken up he was pleasing to God.

Hebrews 11:39 (note) - And all these, having gained approval through their faith, did not receive what was promised

THIS IS THE COVENANT THAT I WILL MAKE WITH THEM AFTER THOSE DAYS SAYS THE LORD: aute e diatheke en diathesomai (1SFMI) pros autous meta tas hemeras ekeinas, legei (3SPAI) kurios: (Heb 8:8-12; Jeremiah 31:33,34; Romans 11:27)

Covenant (1242) (diatheke from dia = two + tithemi = to place pictures that which is placed between two Thus, a covenant is something placed between two, an arrangement between two parties.) was a commonly used in the Greco-Roman world to define a legal transaction in settling an inheritance.  Diatheke denotes an irrevocable decision, which cannot be cancelled by anyone. A prerequisite of its effectiveness before the law is the death of the disposer and thus diatheke was like a "final will and testament".  In reference to the divine covenants, such as the Abrahamic covenant, diatheke is not a covenant in the sense that God came to agreement or compromise with fallen man as if signing a contract. Rather, it involves declaration of God’s unconditional promise to make Abraham and his seed the recipients of certain blessings.

Them = Jews yet this covenant is now available to anyone who believes that Christ's blood is satisfactory & sufficient to save them from the penalty of their sins -- eternal destruction & separation.

The ultimate fulfillment of some of God's promises to Israel in Jer 31:38-40 await complete fulfillment in the millennial age

I WILL PUT MY LAWS UPON THEIR HEART AND UPON THEIR MIND I WILL WRITE THEM: didous (PAPMSN) nomous mou epi kardias auton kai epi ten dianoian auton epigrapso (1SFAI) autous:

God deals with the power of sin by writing the law on our hearts so that we hate sin from the inside and love God's will and walk in his ways freely, not merely by external legal constraint. That's the new covenant = a New Heart 

Ezekiel records God's promises concerning the New Covenant...

Moreover, I will give you a new heart and put a new spirit within you; and I will remove the heart of stone from your flesh and give you a heart of flesh. And I will put My Spirit within you and cause you to walk in My statutes, and you will be careful to observe My ordinances. (Ezekiel 36:26-27)

And I shall give them one heart, and shall put a new spirit within them. And I shall take the heart of stone out of their flesh and give them a heart of flesh, that they may walk in My statutes and keep My ordinances, and do them. Then they will be My people, and I shall be their God. (Ezekiel 11:19-20)

AND THEIR SINS AND THEIR LAWLESS DEEDS I WILL REMEMBER NO MORE: kai ton hamartion auton kai ton anomion auton ou me mnesthesomai (1SFPI) eti:

Remember (3403) (mnaomai through the idea of fixture in mind or of mental grasp) means to recall information from memory, but without necessarily implication persons have actually forgotten

No is actually the double negative (ou me) which gives one of the strongest negatives possible.

Spurgeon in Faith's Checkbook writes the following devotional entitled...

Absolutely No Remembrance: According to this gracious covenant the Lord treats His people as if they had never sinned. Practically, He forgets all their trespasses. Sins of all kinds He treats as if they had never been, as if they were quite erased from His memory. O miracle of grace! God here doth that which in certain aspects is impossible to Him. His mercy worketh miracles which far transcend all other miracles.
      
Our God ignores our sin now that the sacrifice of Jesus has ratified the covenant. We may rejoice in Him without fear that He will be provoked to anger against us because of our iniquities. See! He puts us among the children; He accepts us as righteous; He takes delight in us as if we were perfectly holy. He even puts us into places of trust; makes us guardians of His honor, trustees of the crown jewels, stewards of the gospel. He counts us worthy and gives us a ministry; this is the highest and most special proof that He does not remember our sins. Even when we forgive an enemy, we are very slow to trust him; we judge it to be imprudent so to do. But the Lord forgets our sins and treats us as if we had never erred. O my soul, what a promise is this! Believe it and be happy.

From Our Daily Bread- A Past Long Gone

According to the English novelist Aldous Huxley, "There are no back moves on the chessboard of life." Yet we remain aware of things we have done and things we have left undone. Our sins worry us. They motivate us to wish fervently that somehow we could undo the past.

That's why those who put their faith in Jesus can be thankful for God's message in both the Old and New Testaments. When Paul preached in Antioch, he said, "By [Jesus], everyone who believes is justified from all things from which you could not be justified by the law of Moses" (Acts 13:39). The law condemned us (Romans 7:10-11), but Jesus offers deliverance and new life (8:1).

Are you worried about what you've done in the past? Rejoice! God has "cast all our sins into the depths of the sea" (Micah 7:19). Are you still concerned about your sins? Rejoice! "Their sins and their lawless deeds I will remember no more" (Hebrews 10:17). And "I have blotted out, like a thick cloud, your transgressions" (Isaiah 44:22).

If you have put your faith in Jesus and asked Him to forgive you, the past is truly forgotten. "As far as the east is from the west, so far has He removed our transgressions from us" (Psalm 103:12). Trust and rejoice!—Vernon C Grounds (Our Daily Bread, Copyright RBC Ministries, Grand Rapids, MI. Reprinted by permission. All rights reserved)

NOW WHERE THERE IS FORGIVENESS OF THESE THINGS: hopou de aphesis touton: Mt 26:28

THERE IS NO LONGER ANY OFFERING FOR SIN: ouketi prosphora peri hamartias: