He who is only just is cruel.

Strict justice, applied without any mercy in it, comes out the other side as cruelty. Being within your rights and fair by the rules can still leave a person flattened, and the fairness does not undo that. Rightness is a floor to stand on, not the whole of how to treat someone.

Next time you are clearly in the right, ask what you want to happen after you have proved it. Often a small allowance, extra time or a quiet word instead of a formal one, costs little and changes what the other person can recover from. Keep the standard, and soften the delivery.