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Proof of Regular Level Set Theorem via Inverse FT
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A Level set that doesn't contain critical points must be a regular submanifold, i.e, is locally cut out by coordinate. To show that for the case of F: N to R^m, simply note that at p, once we fix a coordinate patch, the Jacobian of F is a submersion so there must be an mxm invertible submatrix. Use that to replace some coordinates by Fi.
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