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

theorem
  for X1, X2 being open non empty SubSpace of X st X1 meets X2 holds X1
  meet X2 is open SubSpace of X
proof
  let X1, X2 be open non empty SubSpace of X such that
A1: X1 meets X2;
  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 meet X2 as Subset of X by Th1;
  A1 is open & A2 is open by Th16;
  then A1 /\ A2 is open;
  then A is open by A1,Def4;
  hence thesis by Th16;
end;
