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It can be shown that if ''f'' is normal then ''f'' commutes with [[supremum|suprema]]; for any set ''A'' of ordinals,
:''f''(sup ''A'') = sup {''f''(α) : α ∈ ''A'' }.
Indeed, if sup ''A'' is not a limit ordinal then sup ''A'' is an element of ''A'' and the equality follows from the increasing property of ''f''. If sup ''A'' is a limit ordinal then the equality follows from the continuous property of ''f''.
A '''fixed point''' of a normal function is an ordinal β such that ''f''(β) = β.
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