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 Th4:
  [#]GX = A \/ B & A,B are_separated implies
  A is open closed & B is open closed
proof
  assume that
A1: [#]GX = A \/ B and
A2: A,B are_separated;
A3: [#]GX \ B = A by A1,A2,Th1,PRE_TOPC:5;
    B c= Cl B by PRE_TOPC:18;
    then
A4: A misses Cl B implies Cl B = B by A1,XBOOLE_1:73;
    A c= Cl A by PRE_TOPC:18;
    then
A5: Cl A misses B implies Cl A = A by A1,XBOOLE_1:73;
  B = [#]GX \ A by A1,A2,Th1,PRE_TOPC:5;
  hence thesis by A2,A5,A4,A3,PRE_TOPC:22,23;
end;
