What exactly means the expression "either of these assumptions is satisfied"?

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In Analysis II of Amann and Escher I read

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This mean that both conditions must hold or just that any of them (separately or together) must hold to prove the result?

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"Either of these assumptions is satisfied" means "At least one of these assumptions is satisfied". So it could be just the first, it could be just the second, it could be both, and each of these three options is valid.