Equivalence of two statements

How one could prove that these two statements are equivalent:

(1st Statement) For all x in M, there exists B(x, r_x) - open ball which is a subset of R^n, such that (M \cap B(x, r_x)) is a k-dimensional map trace of some patch (patch = local surface) of R^n.
[Where the "\cap" symbol denotes intersection.]

(2nd Statement) For all x in M, there exists B(x, r_x) - open ball which is a subset of R^n, and there exists function f: B(x, r_x) -> R^{n-k}, such that:
(a) rank (d_x f) = n - k (it means: it is maximal),
(b) B(x, r_x) \cap {f = 0} = M \cap B(x, r_x),
[Where the "\cap" symbol denotes intersection and by {f = 0} I mean the set of zeros of function f.]

Chris

PS. I believe that the implication: [1 <= 2] could be somehow proved using Implicit Function Theorem...
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