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1-q
Note that all hypotheses of the theorem are necessary: if the space ''X'' is not complete, no fixed point need exist. Also, the requirement d(''Tx'', ''Ty'') < d(''x'', ''y'') for all ''x'' and ''y'' is not enough to ensure the existence of a fixed point. When using the theorem in practice, the most difficult item to check is typically that ''T'' actually maps elements from ''X'' to ''X'', i.e. that ''Tx'' is always an element of ''X''.
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