theorem Th19:
  for P, A, U holds Polish-atoms(P, A) c= Polish-expression-layer(P, A, U)
proof
  let P, A, U;
  set X = Polish-atoms(P, A);
  set Y = Polish-expression-layer(P, A, U);
  let a;
  assume A1: a in X;
  then reconsider p = a as FinSequence;
  set q = <*>P;
  A3: q in U^^0 by Th4;
  p in P & A.p = 0 by A1, Def7; then
  p^q in Y by A3, Th18;
  hence thesis by FINSEQ_1:34;
end;
