For step 1, observe that for each row of "A" or column of "B", and for each "m"-combination "S", the values of det("AB") and det("A"["m"],"S")det("B""S",["m"]) indeed depend linearly on the row or column. For the latter this is immediate from the multilinear property of the determinant; for the former one must in addition check that taking a linear combination for the row of "A" or column of "B" while leaving the rest unchanged only affects the corresponding row or column of the product "AB", and by the same linear combination. Thus one can work out both sides of the Cauchy−Binet formula by linearity for every row of "A" and then also every column of "B", writing each of the rows and columns as a linear combination of standard basis vectors. The resulting multiple summations are huge, but they have the same form for both sides: corresponding terms involve the same scalar factor (each is a product of entries of "A" and of "B"), and these terms only differ by involving two different expressions in terms of constant matrices of the kind described above, which expressions should be equal according to the Cauchy−Binet formula. This achieves the reduction of the first step.