Use the local normal form of a submersion, f can be written as
. And are the coordinates of the sub-manifold G.
Suppose is another local chart of M and the chart of N with non-empty intersection with the above charts respectively. Since and both are coordinates of N, is only the function of , i.e. it does not depend on . Also is only the function of .
Since M and N are both orientable we can choose the sign of the coordinates so that the Jocobian and , which implies that