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"so he knew killing was a sin"

It translates murder, not killing. God ordered the "death penalty" throughout Old Testament. Worshiping a false god was a capital offense. Paul thought he was being righteous by carrying out God's Law. Here are just a few examples. Saul was probably operating under Leviticus 24:16.

Leviticus 24:16 anyone who blasphemes the name of the LORD must be put to death. The entire assembly must stone him. Whether an alien or native-born, when he blasphemes the Name, he must be put to death.

Exodus 21:16 "Anyone who kidnaps another and either sells him or still has him when he is caught must be put to death.

Exodus 21:17 "Anyone who curses his father or mother must be put to death.

Exodus 31:14 " 'Observe the Sabbath, because it is holy to you. Anyone who desecrates it must be put to death; whoever does any work on that day must be cut off from his people.

Exodus 31:15 "For six days, work is to be done, but the seventh day is a Sabbath of rest, holy to the LORD . Whoever does any work on the Sabbath day must be put to death."

Leviticus 20:27 " 'A man or woman who is a medium or spiritist among you must be put to death. You are to stone them; their blood will be on their own heads.' "