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Paul getting in Peter's face proves they taught the same gospel.
I am not really writing this to you; as addressing you on these issues is pointless....
No, that is not why Paul got in Peter's face.
Rather, he did so because "the righteousness of God WITHOUT the Law" applies to both; Peter had agreed to this, but then had momentarily gone against its' bring true "of the Gentiles also..." (Rom. 3:21-31).
Galatians 2:11 But when Peter was come to Antioch, I withstood him to the face, because he was to be blamed. 2:12 For before that certain came from James, he did eat with the Gentiles: but when they were come, he withdrew and separated himself, fearing them which were of the circumcision. 2:13 And the other Jews dissembled likewise with him; insomuch that Barnabas also was carried away with their dissimulation. 2:14 But when I saw that they walked not uprightly according to the truth of the gospel, I said unto Peter before them all, If thou, being a Jew, livest after the manner of Gentiles, and not as do the Jews, why compellest thou the Gentiles to live as do the Jews?
Note...
2:15 We who are Jews by nature, and not sinners of the Gentiles, 2:16 Knowing that a man is not justified by the works of the law, but by the faith of Jesus Christ, even we have believed in Jesus Christ, that we might be justified by the faith of Christ, and not by the works of the law: for by the works of the law shall no flesh be justified.
Same justification.
But the gospel of God concerning His Son - in other words - the gospel of Christ...is more than justification.
There is that aspect of the gospel of Christ that was Prophesied unto Israel (how that His justification would enable them at last, to KEEP the Law via THEIR PROMISED New Covenant).
And there is that aspect of the gospel of Christ that was kept secret since the world began - until Paul - and that he ALONE preached among the Gentiles.
This aspect was NOT Prophesied and is not by Covenant.
In this, another aspect they nevertheless do share is the role of the Spirit in justification.
Galatians 3:1 O foolish Galatians, who hath bewitched you, that ye should not obey the truth, before whose eyes Jesus Christ hath been evidently set forth, crucified among you? 3:2 This only would I learn of you, Received ye the Spirit by the works of the law, or by the hearing of faith? 3:3 Are ye so foolish? having begun in the Spirit, are ye now made perfect by the flesh?
3:6 Even as Abraham believed God, and it was accounted to him for righteousness.
And that is referring to Abraham back WHEN he WAS in UN-Circumcision.
It is NOT referring to Abraham LATER - AFTER he was in Circumcision and became Isaac's father, out of whom Jacob sprang (out of whom sprang, in turn, Israel's Twelve Patriarchs, et al).
Romans 4:9 Cometh this blessedness then upon the circumcision only, or upon the uncircumcision also? for we say that faith was reckoned to Abraham for righteousness. 4:10 How was it then reckoned? when he was in circumcision, or in uncircumcision? Not in circumcision, but in uncircumcision. 4:11 And he received the sign of circumcision, a seal of the righteousness of the faith which he had yet being uncircumcised: that he might be the father of all them that believe, though they be not circumcised; that righteousness might be imputed unto them also: 4:12 And the father of circumcision to them who are not of the circumcision only, but who also walk in the steps of that faith of our father Abraham, which he had being yet uncircumcised. 4:13 For the promise, that he should be the heir of the world, was not to Abraham, or to his seed, through the law, but through the righteousness of faith. 4:14 For if they which are of the law be heirs, faith is made void, and the promise made of none effect: 4:15 Because the law worketh wrath: for where no law is, there is no transgression. 4:16 Therefore it is of faith, that it might be by grace; to the end the promise might be sure to all the seed; not to that only which is of the law, but to that also which is of the faith of Abraham; who is the father of us all, 4:17 (As it is written, I have made thee a father of many nations,) before him whom he believed, even God, who quickeneth the dead, and calleth those things which be not as though they were.