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The relevant rule here is R4-4a. Decision 4-4a/12 is also relevant.
You should have added a 2 shot penalty to your score for the hole for using your partner’s club meaning that you would have scored 7 for 0 points. Therefore your partner's score was the better score for the hole.
You have been penalised twice because you incorrectly assumed that once you had used your partners club he may no longer use it whilst in fact he may.