Finite embedding theorems for partial Steiner triple systems

Let P be a finite set and (P, 1), (P, 2),…, (P, k) any collection of mutually disjoint partial Steiner triple systems. Then these partial triple systems can be embedded in finite mutually disjoint triple systems (S, 1), (S, 2),…, (S, k). This result is then used to prove the following more general result. If (P, 1), (P, 2),…, (P, k) are any collection of finite partial Steiner triple systems, then these partial triple systems can be embedded in finite triple systems (S, 1), (S, j = i ∩ j for all i ≠ j = 1, 2,…, k.