reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;

theorem Th16:
  for P1, P2, Q1, Q2 st P1 c= P2 & Q1 c= Q2 holds P1^Q1 c= P2^Q2
proof
  let P1, P2, Q1, Q2;
  assume A1: P1 c= P2 & Q1 c= Q2;
  let a;
  assume a in P1^Q1;
  then consider p, q such that A3: a = p^q & p in P1 & q in Q1 by Def2;
  thus thesis by A1, A3, Def2;
end;
