and through this he exists mediator of a new covenant, so that the ones called receive the promised eternal inheritance. A death has occurred to the extent of redemption of the transgressions under the first covenant (Hebrews 9:15)
... in him also believing sealed of the promised Holy Spirit who exists the guarantor of our inheritance, to the extent redemption of his possession, to the praise of His glory! (Ephesian 1:13-14)
Earlier Paul writes we were chose in love and "predestined for adoption to the extent of His own through Jesus Christ" (Ephesians 1:4-5). We are made His own, his children through adoption of the Spirit. The concept of adoption carries with that we are born of God. Jesus said this is how we enter the kingdom of God. Predestination occurs through adoption.
...sealed of the promised Holy Spirit who exists the guarantor of our inheritance...
Jesus promised that he would send the Holy Spirit to be with us. So it is the promised Holy Spirit in that sense but to a greater extent the Spirit is how God brings about the promise of eternal inheritance, of us being heirs, sons. This God promised in the Old Testament an eternal covenant and inheritance (Genesis 17:7; Jeremiah 31, 32; 32:40; Psalm 105:10; Psalm 111:9; Hosea 2:19; Luke 1:67-79; Galatians 3:29; Galatians 3:16).
In whom obtaining an inheritance, predestined according to the purpose of him, the one working all according to the counsel of his will, so that we who were the first to hope in Christ might be to the praise of his glory (Ephesians 1:11-14).
We obtain an inheritance through predestination, as Paul writes this was before he created the world, he chose us and predestined us. And he seals as his own and guarantees to take possession of his own, to the praise his glory, "sealed of the promised Holy Spirit who exists the guarantor of our inheritance, to the extent redemption of his possession, to the praise of His glory!" Those who are predestined God is working out all, our glorification, to the praise of his glory.
according to the eternal purpose that which he did in Christ Jesus our Lord (Ephesians 3:11)
An eternal inheritance is to those he predestined and called, this is what God eternally purposed in Christ. This manifold wisdom of God through the church is being made known, this insight into what God has eternally purposed in Christ (Ephesians 3:10). An eternal inheritance along with eternal redemption, is being made known, according to the counsel of his will and to the praise of his glory.
now, the God of peace, the one in the blood covenant, the eternal covenant, brought up from the dead the great Shepherd of the sheep, the one our Lord Jesus (Hebrews 13:20)
The new covenant is a eternal covenant a covenant in his blood (Matthew 26:28). Notice the connection between the blood and the Spirit, he shed his blood and was raised in Spirit. This is why he shed his blood to establish a new covenant. "For where there is a covenant it requires that a death be made to establish" (Hebrews 9:16). A covenant is a will, a testament. He entered once into the Holy Place, "through his own blood obtaining eternal redemption" (Hebrews 9:12-15). Jesus died establishing eternal redemption. It was through the Spirit he was raised and through the Spirit he offered himself without blemish to God (Hebrews 9:14). Thus in this eternal covenant we have received blood bought eternal redemption and been made alive through the eternal Spirit, adopted as his own, predestined and called to receive the promised eternal inheritance.
for if the blood of goats and bulls and cleansing ashes of heifers sanctify the one defiled from, that the flesh is purified, how much more the blood of Jesus whom through the eternal Spirit offered himself without blemish to God, purify that our conscious from dead works to worship a living God! (Hebrews 9:14)
There was a purpose in the sacrificing of animals in the old covenant; the cleansing of the defiled flesh. God had chosen Israel as his own, but still there was a sacrifice for sin done in the body. But it was temporal and did not take away sin eternally, thus had to be offered each year. Jesus's offering of his blood took away sin, securing eternal redemption (1 John 3:5, John 1:29; ; Hebrews 9:26; 1 John 2:2; 1 peter 1:18-19; Isaiah 44:22; 1 Corinthians 6:20; Titus 2:14; Ephesians 1:7).
In the eternal covenant we receive eternal redemption through his blood and an eternal inheritance through adoption of the Spirit. Jesus takes away sin, in the sense he is a propitiation for sin, once and for all. And through the blood bought redemption we are redeemed from sin's debt. And Jesus takes away sin in that the new covenant gives us the means by which we rule over sin, and purify the defiled flesh so that we have a good conscious toward God.
"Keep your eyes on me, I have good things for you"