reserve GX for TopSpace;
reserve A, B, C for Subset of GX;
reserve TS for TopStruct;
reserve K, K1, L, L1 for Subset of TS;

theorem Th9:
  K is closed & L is closed or K is open & L is open implies
  K \ L, L \ K are_separated
proof
A1: now
    let K,L be Subset of TS such that
A2: K is open and
A3: L is open;
A4: K /\ L` c= K by XBOOLE_1:17;
A5: (Cl L) /\ K` c= K` by XBOOLE_1:17;
    Cl([#]TS \ K) = [#]TS \ K by A2,PRE_TOPC:23;
    then
A6: (K /\ (L`)) /\ ((Cl L) /\ Cl(K`)) c= K /\ (K`) by A4,A5,XBOOLE_1:27;
A7: K \ L = K /\ (L`) by SUBSET_1:13;
A8: L /\ K` c= L by XBOOLE_1:17;
    (Cl K) /\ (L`) c= L` by XBOOLE_1:17;
    then
A9: ((Cl K) /\ L`) /\ (L /\ K`) c= L /\ (L`) by A8,XBOOLE_1:27;
    L misses L` by XBOOLE_1:79;
    then
A10: L /\ L` = {};
    K misses K` by XBOOLE_1:79;
    then
A11: K /\ K` = {};
    [#]TS \ K = K`;
    then
A12: ((Cl K) /\ Cl(L`)) /\ (L /\ (K`)) = ((Cl K) /\ (L`)) /\ (L /\ (K`))
    by A3,PRE_TOPC:23;
A13: L \ K = L /\ (K`) by SUBSET_1:13;
    Cl(L /\ (K`)) c= (Cl L) /\ Cl(K`) by PRE_TOPC:21;
    then (K \ L) /\ Cl(L \ K) c= (K /\ (L`)) /\ ((Cl L) /\ Cl(K`)) by A7,A13,
XBOOLE_1:27;
    then (K \ L) /\ Cl(L \ K) = {}TS by A11,A6;
    then
A14: (K \ L) misses Cl(L \ K);
    Cl(K /\ (L`)) c= (Cl K) /\ Cl (L`) by PRE_TOPC:21;
    then
    (Cl(K \ L)) /\ (L \ K) c= ((Cl K) /\ Cl(L`)) /\ (L /\ (K`)) by A7,A13,
XBOOLE_1:27;
    then (Cl(K \ L)) /\ (L \ K) = {}TS by A12,A10,A9;
    then (Cl(K \ L)) misses (L \ K);
    hence K \ L,L \ K are_separated by A14;
  end;
A15: now
    let K,L be Subset of TS;
    assume that
A16: K is closed and
A17: L is closed;
A18: [#]TS \ L is open by A17,PRE_TOPC:def 3;
A19: ([#]TS \ K) \ ([#]TS \ L) = (K`) /\ (([#]TS \ L)`) by SUBSET_1:13
      .= (K`) /\ L by PRE_TOPC:3
      .= L \ K by SUBSET_1:13;
A20: ([#]TS \ L) \ ([#]TS \ K) = (L`) /\ (([#]TS \ K)`) by SUBSET_1:13
      .= (L`) /\ K by PRE_TOPC:3
      .= K \ L by SUBSET_1:13;
    [#]TS \ K is open by A16,PRE_TOPC:def 3;
    hence K \ L,L \ K are_separated by A1,A18,A20,A19;
  end;
  assume K is closed & L is closed or K is open & L is open;
  hence thesis by A1,A15;
end;
