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, X2 being T_0 non empty SubSpace of X holds X1 is open or X2 is
  open implies X1 union X2 is T_0
proof
  let X1, X2 be T_0 non empty SubSpace of X;
  reconsider A1 = the carrier of X1, A2 = the carrier of X2 as non empty
  Subset of X by TSEP_1:1;
  assume X1 is open or X2 is open;
  then
A1: A1 is open or A2 is open by TSEP_1:16;
  A1 is T_0 & A2 is T_0 by Th13;
  then the carrier of X1 union X2 = A1 \/ A2 & A1 \/ A2 is T_0 by A1,Th7,
TSEP_1:def 2;
  hence thesis;
end;
