LPT: If you catch someone in a lie, and they still refuse it, leave the conversation immediately
If someone is lying through their teeth over something they are clearly guilty of, it's not worth your time. You can't use reason to get someone out of an unreasonable position. Arguing further just encites more emotions, breeds further conflict, and is a waste of time. It makes a resolution more difficult for the both of you. Cut them off or wait until emotions have settled to continue.