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;
