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 Th34:
  ((X1 union X2) meets X0 iff X1 meets X0 or X2 meets X0) & (X0
  meets (X1 union X2) iff X0 meets X1 or 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;
A1: X1 meets X0 or X2 meets X0 implies (X1 union X2) meets X0
  proof
    assume X1 meets X0 or X2 meets X0;
    then A1 meets A0 or A2 meets A0 by TSEP_1:def 3;
    then (A1 /\ A0) \/ (A2 /\ A0) <> {};
    then (A1 \/ A2) /\ A0 <> {} by XBOOLE_1:23;
    then (the carrier of (X1 union X2)) meets A0 by TSEP_1:def 2;
    hence thesis by TSEP_1:def 3;
  end;
A2: (X1 union X2) meets X0 implies X1 meets X0 or X2 meets X0
  proof
    assume (X1 union X2) meets X0;
    then (the carrier of (X1 union X2)) meets A0 by TSEP_1:def 3;
    then (the carrier of (X1 union X2)) /\ A0 <> {};
    then (A1 \/ A2) /\ A0 <> {} by TSEP_1:def 2;
    then (A1 /\ A0) \/ (A2 /\ A0) <> {} by XBOOLE_1:23;
    then A1 meets A0 or A2 meets A0;
    hence thesis by TSEP_1:def 3;
  end;
  hence (X1 union X2) meets X0 iff X1 meets X0 or X2 meets X0 by A1;
  thus thesis by A2,A1;
end;
