reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th27:
  for D being non empty a_partition of the carrier of X, A being
Subset of D holds union A in the topology of X iff A in the topology of space D
proof
  let D be non empty a_partition of the carrier of X, B be Subset of D;
A1: the topology of space D = { A where A is Subset of D : union A in the
  topology of X } by Def7;
  hence union B in the topology of X implies B in the topology of space D;
  assume B in the topology of space D;
  then ex A being Subset of D st B = A & union A in the topology of X by A1;
  hence thesis;
end;
