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 Th3:
  [#]GX = A \/ B & A is open & B is open & A misses B implies A,B are_separated
proof
  assume that
A1: [#]GX = A \/ B and
A2: A is open and
A3: B is open and
A4: A misses B;
A5: Cl([#]GX \ B) = [#]GX \ B by A3,PRE_TOPC:23;
  A = [#]GX \ B by A1,A4,PRE_TOPC:5;
  then
A6: A is closed by A5,PRE_TOPC:22;
A7: Cl([#]GX \ A) = [#]GX \ A by A2,PRE_TOPC:23;
  B = [#]GX \ A by A1,A4,PRE_TOPC:5;
  then B is closed by A7,PRE_TOPC:22;
  hence thesis by A4,A6,Th2;
end;
