theorem
  P is nowhere_dense & Q is nowhere_dense implies P \/ Q is nowhere_dense
proof
  assume that
A1: P is nowhere_dense and
A2: Q is nowhere_dense;
A3: Cl Q is boundary by A2;
  Cl P is boundary by A1;
  then Cl P \/ Cl Q is boundary by A3,Th49;
  then Cl(P \/ Q) is boundary by PRE_TOPC:20;
  hence thesis;
end;
