Smooth homotopy theorem

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  1. Suppose f:M \to N is a continuous map between differential manifolds. Then, there exists a smooth map f':M \to N such that f' is homotopy-equivalent to f
  2. Suppose F:M \times I \to N is a homotopy between smooth maps f_0:M \to N and f_1:M \to N. Then, there exists a homotopy F':M \times I \to N from f_0 to f_1 that is smooth as a map from M \times I to N.