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 Th8:
  A,B are_separated & A,C are_separated implies A,B \/ C are_separated
proof
  assume that
A1: (Cl A) misses B and
A2: A misses Cl B and
A3: (Cl A) misses C and
A4: A misses Cl C;
A5: A /\ Cl B = {} by A2;
  A /\ Cl (B \/ C) = A /\ ((Cl B) \/ Cl C) by PRE_TOPC:20
    .= (A /\ Cl B) \/ (A /\ Cl C) by XBOOLE_1:23
    .= {}GX by A4,A5;
  then
A6: A misses Cl (B \/ C);
A7: (Cl A) /\ B = {} by A1;
  (Cl A) /\ (B \/ C) = ((Cl A) /\ B) \/ ((Cl A) /\ C) by XBOOLE_1:23
    .= {}GX by A3,A7;
  then (Cl A) misses (B \/ C);
  hence thesis by A6;
end;
