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 implies ((X1 meet X2) meets X0 implies X1 meets X0 & X2
  meets X0) & (X0 meets (X1 meet X2) implies X0 meets X1 & X0 meets X2)
proof
  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 A2 = the carrier of X2 as Subset of X by TSEP_1:1;
  assume
A1: X1 meets X2;
  thus (X1 meet X2) meets X0 implies X1 meets X0 & X2 meets X0
  proof
    assume (X1 meet X2) meets X0;
    then (the carrier of (X1 meet X2)) meets A0 by TSEP_1:def 3;
    then (A1 /\ A2) /\ A0 <> {} by A1,TSEP_1:def 4;
    then
A2: A1 /\ (A2 /\ (A0 /\ A0)) <> {} by XBOOLE_1:16;
    then A1 /\ (A0 /\ (A2 /\ A0)) <> {} by XBOOLE_1:16;
    then (A1 /\ A0) /\ (A2 /\ A0) <> {} by XBOOLE_1:16;
    then
A3: A1 meets A0;
    A2 /\ A0 <> {} by A2;
    then A2 meets A0;
    hence thesis by A3,TSEP_1:def 3;
  end;
  hence thesis;
end;
