reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem Th38:
  for X0 being closed non empty SubSpace of X holds X0 meets X1
  implies X0 meet X1 is closed SubSpace of X1
proof
  let X0 be closed non empty SubSpace of X;
  reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
  reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
  reconsider B = A0 /\ A1 as Subset of X1 by XBOOLE_1:17;
  B = A0 /\ [#]X1 & A0 is closed by TSEP_1:11;
  then
A1: B is closed by PRE_TOPC:13;
  assume
A2: X0 meets X1;
  then B = the carrier of X0 meet X1 by TSEP_1:def 4;
  hence thesis by A2,A1,TSEP_1:11,27;
end;
