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 Th12:
  for P, p, q st p in P* & q in P* holds p^q in P*
proof
  let P, p, q;
  assume that A1: p in P* and A2: q in P*;
  consider m such that A3: p in P^^m by A1, Th5;
  consider n such that A4: q in P^^n by A2, Th5;
  p^q in P^^(m+n) by Th11, A3, A4;
  hence thesis by Th5;
end;
