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You have to use the trademark at least in class x. You may use the mark in other categories as well but the mark would generally not be protected under trademark law because the use is outside the registered use
In conclusion, one is allowed to use the mark in other uses just notes that there will not be trademark protection without registration in those classes.
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If they don't have a trademark in class Y then matters become somewhat murky and require a factual examination by a court/jury whether they have might have usage rights arising under common law trademark rights. Typically common law trademark protection extend intra-state while Federal trademark protection extends nationwide.
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