Laplace Expansion

For simplicity, we follow the rules: x, y are sets, N is an element of N, c, i, j, k, m, n are natural numbers, D is a non empty set, s is an element of 2Set Seg(n + 2), p is an element of the permutations of n-element set, p1, q1 are elements of the permutations of (n+ 1)-element set, p2 is an element of the permutations of (n+ 2)-element set, K is a field, a, b are elements of K, f is a finite sequence of elements of K, A is a matrix over K, A1 is a matrix over D of dimension n × m, p3 is a finite sequence of elements of D, and M is a matrix over K of dimension n. The following propositions are true: (1) For every finite sequence f and for every natural number i such that i ∈ dom f holds len(f i) = len f −′ 1. (2) Let i, j, n be natural numbers and M be a matrix over K of dimension n. If i ∈ domM, then len (the deleting of i-row and j-column in M) = n−′ 1. (3) If j ∈ Seg widthA, then width (the deleting of j-column in A) = widthA−′ 1.