integrality has a specialized use in mathematics. As a "normal" word it means entireness or completeness, and it is very, very rarely used, though the adjectival form integral is used less rarely. integrity is much more often used.
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bbserI admire her artistic integrity.What's this mean? She has a lot of common skills in art?No. She adheres firmly to certain artistic values. For example, she only produces art of the highest quality, and cannot be persuaded to produce art of lower quality just to make money.