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 union X2) misses X0 iff X1 misses X0 & X2 misses X0) & (X0 misses
  (X1 union X2) iff X0 misses X1 & X0 misses 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 union X2) misses X0 implies X1 misses X0 & X2 misses X0
  proof
    assume (X1 union X2) misses X0;
    then (the carrier of (X1 union X2)) misses A0 by TSEP_1:def 3;
    then (A1 \/ A2) /\ A0 = {} by TSEP_1:def 2;
    then
A2: (A1 /\ A0) \/ (A2 /\ A0) = {} by XBOOLE_1:23;
    then
A3: A2 misses A0;
    A1 /\ A0 = {} by A2;
    then A1 misses A0;
    hence thesis by A3,TSEP_1:def 3;
  end;
A4: X1 misses X0 & X2 misses X0 implies (X1 union X2) misses X0
  proof
    assume that
A5: X1 misses X0 and
A6: X2 misses X0;
    A1 misses A0 by A5,TSEP_1:def 3;
    then
A7: A1 /\ A0 = {};
    A2 misses A0 by A6,TSEP_1:def 3;
    then (A1 /\ A0) \/ (A2 /\ A0) = {} by A7;
    then (A1 \/ A2) /\ A0 = {} by XBOOLE_1:23;
    then (the carrier of (X1 union X2)) /\ A0 = {} by TSEP_1:def 2;
    then (the carrier of (X1 union X2)) misses A0;
    hence thesis by TSEP_1:def 3;
  end;
  hence (X1 union X2) misses X0 iff X1 misses X0 & X2 misses X0 by A1;
  thus thesis by A1,A4;
end;
