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
  X1 meets X2 & X1 is SubSpace of X0 & X2 is SubSpace of X0 implies X1
  meet X2 is SubSpace of X0
proof
  assume
A1: X1 meets X2;
  assume X1 is SubSpace of X0 & X2 is SubSpace of X0;
  then the carrier of X1 c= the carrier of X0 & the carrier of X2 c= the
  carrier of X0 by TSEP_1:4;
  then
A2: (the carrier of X1) \/ (the carrier of X2) c= the carrier of X0 by
XBOOLE_1:8;
  (the carrier of X1) /\ (the carrier of X2) c= (the carrier of X1) \/ (
  the carrier of X2) by XBOOLE_1:29;
  then (the carrier of X1) /\ (the carrier of X2) c= the carrier of X0 by A2,
XBOOLE_1:1;
  then the carrier of (X1 meet X2) c= the carrier of X0 by A1,TSEP_1:def 4;
  hence thesis by TSEP_1:4;
end;
