reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;

theorem
  for X1 being T_0 non empty SubSpace of X, X2 being non empty SubSpace
  of X holds X1 meets X2 implies X1 meet X2 is T_0
proof
  let X1 be T_0 non empty SubSpace of X, X2 be non empty SubSpace of X;
  reconsider A1 = the carrier of X1 as non empty Subset of X by TSEP_1:1;
  reconsider A2 = the carrier of X2 as non empty Subset of X by TSEP_1:1;
  assume X1 meets X2;
  then
A1: the carrier of X1 meet X2 = A1 /\ A2 by TSEP_1:def 4;
  A1 is T_0 by Th13;
  then A1 /\ A2 is T_0 by Th6;
  hence thesis by A1;
end;
