Suppose that V is open in Rn, that a V, and that F : V R is C1 on V. If F(a) = 0 Fxj(a)

Suppose that V is open in Rn, that a ˆˆ V, and that F : V †’ R is C1 on V. If F(a) = 0 ‰  Fxj(a) and u(j): = (x1 . . . . .. . . .-xj-1, xj+1,. . . . . . . . ., xn) for j = 1,2, ...,n, prove that there exist open sets Wj containing (a1,. . . . . . . . aj, aj+1, . . . . . . , an), an r > 0, and functions gj(u(j)), C1 on Wj, such that F(x1,. . . . . . . . . xj-1, gj(u(j)), xj+1,. . . . . . . . . xn) = 0 on Wj and

On Br(a).

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