What do you mean by the "other" Paul?

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This means that sin and death for everyone came from (originated, or resulted from the actions of) one man, Adam, Romans 5:12 (WEB):
(12) Therefore as sin entered into the world through one man, and death through sin; so death passed to all men, because all sinned.
and the resurrection of the dead also came from (orginated, or resulted from the actions of) one man, Jesus, the Christ, Romans 5:15-21 (WEB):
(15) But the free gift isn’t like the trespass. For if by the trespass of the one the many died, much more did the grace of God, and the gift by the grace of the one man, Jesus Christ, abound to the many.
(16) The gift is not as through one who sinned: for the judgment came by one to condemnation, but the free gift came of many trespasses to justification.
(17) For if by the trespass of the one, death reigned through the one; so much more will those who receive the abundance of grace and of the gift of righteousness reign in life through the one, Jesus Christ.
(18) So then as through one trespass, all men were condemned; even so through one act of righteousness, all men were justified to life.
(19) For as through the one man’s disobedience many were made sinners, even so through the obedience of the one [man], many will be made righteous.
(20) The law came in besides, that the trespass might abound; but where sin abounded, grace abounded more exceedingly;
(21) that as sin reigned in death, even so grace might reign through righteousness to eternal life through Jesus Christ our Lord.
It was through the acts of Jesus, while he was a man (100% man), that we are saved and redeemed, and can look forward to the promised resurrection to eternal life.
Paul is saying that God has appointed judgement of the world to the man Jesus, who He has raised from the dead. It is evident that since his resurrection Jesus is no longer a man, for he is not living on the Earth and we are told that he is sat down in heaven at the right hand of God.
1 Cornthians 15:45-49 (WEB):
(45) So also it is written, “The first man, Adam, became a living soul.” The last Adam became a life-giving spirit.
(46) However that which is spiritual isn’t first, but that which is natural, then that which is spiritual.
(47) The first man is of the earth, made of dust. The second man is the Lord from heaven.
(48) As is the one made of dust, such are those who are also made of dust; and as is the heavenly, such are they also that are heavenly.
(49) As we have borne the image of those made of dust, let’s also bear the image of the heavenly.
Irrelevant to Jesus' nature.
Why is the man Jesus the mediator between men and God? Because, continuing the sentence to verse 6 (KJV):
(6) Who gave himself a ransom for all, to be testified in due time.
It was Jesus as a man who sacrificed his life for all mankind, and as a man was familiar with our trials and temptations, Hebrews 4:15 (WEB):
(15) For we don’t have a high priest who can’t be touched with the feeling of our infirmities, but one who has been in all points tempted like we are, yet without sin.
Paul is explaining that Jesus' ministry as a high priest is more excellent than the ministry of human Aaronic priesthood, because he is the "mediator of a better covenant, which has been enacted on better promises", and the blood of the sacrifice of his human life was far greater than the blood of sacrifice of animals, which were a type pointing foward to Jesus' sacrifice.
Hebrews 5:4-10 (WEB):
(4) Nobody takes this honor [of being high priest] on himself, but he is called by God, just like Aaron was.
(5) So also Christ didn’t glorify himself to be made a high priest, but it was he who said to him, “You are my Son. Today I have become your father.”
(6) As he says also in another place, “You are a priest forever, after the order of Melchizedek.”
(7) He, in the days of his flesh, having offered up prayers and petitions with strong crying and tears to him who was able to save him from death, and having been heard for his godly fear,
(8) though he was a Son, yet learned obedience by the things which he suffered.
(9) Having been made perfect, he became to all of those who obey him the author of eternal salvation,
(10) named by God a high priest after the order of Melchizedek.
Why would Paul say "in the days of his flesh"? Because Jesus is no longer in the days of his flesh, he has been perfected, glorified and been made a spirit being. Jesus prayed to God, "And now, O Father, glorify thou me with thine own self with the glory which I had with thee before the world was" (John 17:5). When he existed with God before the world had been made, no human had been made, and so obviously Jesus was not a man then. So Jesus was praying for God to restore him to his previous spirit being form after he had sacrificed his human body. God not only answered that prayer but he highly exalted him, Philippians 2:8-11 (WEB):
(8) And being found in fashion as a man, he humbled himself, and became obedient unto death, even the death of the cross.
(9) Wherefore God also hath highly exalted him, and given him a name which is above every name:
(10) That at the name of Jesus every knee should bow, of things in heaven, and things in earth, and things under the earth;
(11) And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.
We (Christians) have the promise of being changed to be made to be like Jesus is now (1 John 3:2), having an immortal (cannot possibly die) spirit nature ("It is sown a natural body; it is raised a spiritual body", 1 Corinthians 15:44), the divine nature (2 Peter 1:4). If that's how we'll be then it is clear that Jesus is no longer a man.
2 Corinthians5:16-17 (WEB):
(16) Therefore we know no one after the flesh from now on. Even though we have known Christ after the flesh, yet now we know him so no more.
(17) Therefore if anyone is in Christ, he is a new creation. The old things have passed away. Behold, all things have become new.