Ky Fan lemma

and L a boundary-odd labeling of T. If L has no complementary edge, then L has an odd number of n-dimensional alternating simplices.

By definition, an n-dimensional alternating simplex must have labels with n + 1 different sizes.

KFL can be proved constructively based on a path-based algorithm.

The algorithm it starts at a certain point or edge of the triangulation, then goes from simplex to simplex according to prescribed rules, until it is not possible to proceed any more.

It can be proved that the path must end in an alternating simplex.

At some edge e, the labeling must change from negative to positive.

Since L has no complementary edges, e must have a negative label and a positive label with a different size (e.g. −1 and +2); this means that e is a 1-dimensional alternating simplex.

Moreover, if at any point the labeling changes again from positive to negative, then this change makes a second alternating simplex, and by the same reasoning as before there must be a third alternating simplex later.

Hence, the number of alternating simplices is odd.

The following description illustrates the induction step for

By the induction basis, this interval must have an alternating simplex, e.g. an edge with labels (+1,−2).

Moreover, the number of such edges on both intervals is odd.

Using the boundary criterion, on the boundary we have an odd number of edges where the smaller number is positive and the larger negative, and an odd number of edges where the smaller number is negative and the larger positive.

By induction, this proof can be extended to any dimension.