reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;

theorem
  for X1, X2 being closed non empty SubSpace of X holds X1 union X2 is
  closed SubSpace of X
proof
  let X1, X2 be closed non empty SubSpace of X;
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
  reconsider A = the carrier of X1 union X2 as Subset of X by Th1;
  A1 is closed & A2 is closed by Th11;
  then A1 \/ A2 is closed;
  then A is closed by Def2;
  hence thesis by Th11;
end;
