A proof of Tutte's realizability condition

This paper gives a simple proof of Tutte's realizability condition for a cutset (circuit) matrix of a nonoriented graph [1],[2]. First, a minimum nonrealizable matrix is defined as~a matrix
[ N U]
that satisfies 1)
[N U]
is not a cutset (circuit) matrix, 2)
[ N U]
does not satisfy the conditions in Tutte's theorem, and 3) deleting any column of
N1
or any row of any normal form
[N1 U]
of
[N U]
, the resultant matrix is realizable as a cutset (circuit) matrix. A proof of Tutte's theorem in this paper is accomplished by showing that minimum nonrealizable matrices do not exist.