structural:reformatting, structural:summarization
Partial proof

For the case of perfect fractional matchings, both the above theorems can derived from the colorful Caratheodory theorem.

For every edge e in E, let 1 be a vector of size | V |, where for each vertex v in V, element v in 1 equals 1 if e is adjacent to v, and 0 otherwise (so each vector 1 has 2 ones and | V | -2 zeros). Every fractional matching corresponds to a conical combination of edges, in which each element is at most 1. A conical combination in which each element is exactly 1 corresponds to a perfect fractional matching. In other words, a collection F of edges admits a perfect fractional matching, if and only if 1 (the vector of | V | ones) is contained in the conical hull of the vectors 1 for e in F.

Consider a graph with 2 n vertices, and suppose there are 2 n subsets of edges, each of which admits a perfect fractional matching (of size n). This means that the vector 1 is in the conical hull of each of these n subsets. By the colorful Caratheodory theorem, there exists a selection of 2 n edges, one from each subset, that their conical hull contains 1. This corresponds to a rainbow perfect fractional matching. The expression 2 n is the dimension of the vectors 1 - each vector has 2 n elements.

Now, suppose that the graph is bipartite. In a bipartite graph, there is a constraint on the vectors 1: the sum of elements corresponding to each part of the graph must be 1. Therefore, the vectors 1 live in a (2 n – 1) -dimensional space. Therefore, the same argument as above holds when there are only 2 n – 1 subsets of edges.