Let F be a free abelian group with basis{a₁, ... , a_m). Let K be the subgroup of F generated by b₁ = r₁₁a₁ + • "• + r_1ma_m ... , ... , b_n = r_n1a₁ + • • • + r_nma_m (r_ij ϵ Z).

(a) For each i, both {b1, ..• , b_i-1,-b_i,b_i+1, . .. , bn} and {b₁, ... , b_i-1,b_i + rb_j, b_i+1, ... , bn} (r ϵ Z; i ≠ j) generate K.

(b) For each i {a₁, ... , a_i-1,-a_i,a_i+1, •.• , a_n} is a basis of F relative to which bj = r_j1a₁ + · · · + r_ji-1a_i-1 - r_jt(-a_i) + r_j,i+₁a_i+1 + :. · + r_jma_m.

(c) For each i and j ≠ i {a₁, ... , a_j-1,a_j - ra_i,,a_j+1, . .. , a_m} (r ϵ Z) is a basis of F relative to which b_k = r_k1a₁ + · · · + r_k,i-1a_i-1 + (r_ki + rr_kj)a_i + r_k,i+1a_i+1 + . " "+ r_k.j-1a_j-l + r_ki(a_j - ra_i) + r_k.j+1a_j+1,+l + • · • + r_kma_m•