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| I_think_you_mean |
The king and I went to the library.
She waved at him and me.
Working out how you use 'I' and 'me' in a sentence where you're describing something you did with another person (or something that was done to you by another person) is actually quite easy
Say the sentence without the other person in it... So, neither 'Me' went to the library' or 'She waved at I' quite work. 
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It is also usual to put yourself second in the sentence - just out of politeness. But then you might be a Dom/me.
