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 Th16:
  A is connected & A c= B \/ C & B,C are_separated implies A c= B or A c= C
proof
  assume that
A1: A is connected and
A2: A c= B \/ C and
A3: B,C are_separated;
A4: A /\ C c= C by XBOOLE_1:17;
  A /\ B c= B by XBOOLE_1:17;
  then
A5: A /\ B,A /\ C are_separated by A3,A4,Th7;
A6: (A /\ B) \/ (A /\ C) = A /\ (B \/ C) by XBOOLE_1:23
    .= A by A2,XBOOLE_1:28;
  assume that
A7: not A c= B and
A8: not A c= C;
  A meets C by A2,A7,XBOOLE_1:73;
  then
A9: A /\ C <> {};
  A meets B by A2,A8,XBOOLE_1:73;
  then
A10: A /\ B <> {};
  then A <> {}GX;
  hence contradiction by A1,A10,A9,A5,A6,Th15;
end;
