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 misses X0 or X2 misses X0 implies (X1 meet X2)
  misses X0) & (X0 misses X1 or X0 misses X2 implies X0 misses (X1 meet 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 misses X0 or X2 misses X0 implies (X1 meet X2) misses X0
  proof
    assume X1 misses X0 or X2 misses X0;
    then A1 misses A0 or A2 misses A0 by TSEP_1:def 3;
    then (A1 /\ A0) /\ (A2 /\ A0) = {};
    then A1 /\ ((A2 /\ A0) /\ A0) = {} by XBOOLE_1:16;
    then A1 /\ (A2 /\ (A0 /\ A0)) = {} by XBOOLE_1:16;
    then (A1 /\ A2) /\ A0 = {} by XBOOLE_1:16;
    then (the carrier of (X1 meet X2)) /\ A0 = {} by A1,TSEP_1:def 4;
    then (the carrier of (X1 meet X2)) misses A0;
    hence thesis by TSEP_1:def 3;
  end;
  hence thesis;
end;
