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1. F(x_0,y_0,z_0)=0

2. F has continuous partial derivatives in M_0 and around it

3. F'_z(x_0,y_0,z_0)=0

4. gradF at (x_0,y_0,z_0) != 0

5. It is known that there is a function f(x,y) so that F(x,y,f(x,y)) =0 in M_0 and around it

Prove that f(x,y) is differentiable in (x_0, y_0)