theorem Th17:
  for A being Subset of TOP-REAL n holds UBD A c= A`
proof
  let A be Subset of TOP-REAL n;
  reconsider D=UBD A as Subset of (TOP-REAL n) | A` by Th12;
  D c= the carrier of ((TOP-REAL n) | A`);
  hence thesis by PRE_TOPC:8;
end;
