It is worth noting that the above proposition proves that if , then is smooth in the neighborhood of . In other words, knowledge concerning at one point can sometimes yield conclusions in the neighborhood of that point; see [79, Remark 1.6, p. 985].
Equation (48) essentially has the same content as Equation (38). In order to see this, define the object inside the norm on the left-hand side of Equation (48) to be . Then
Using the above expansions and equations, expressions for the higher-order time derivatives can be derived.
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