  We establish Pfaffian analogues of the Cauchy--Binet formula and the
Ishikawa--Wakayama minor-summation formula. Each of these Pfaffian analogues
expresses a sum of products of subpfaffians of two skew-symmetric matrices in
terms of a single Pfaffian. By using these Pfaffian formulas we give new
transparent proofs to several identities for Schur Q23 pa-functions.
