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 Th12:
  GX is connected iff for A being Subset of GX st A <> {}GX & A <>
  [#]GX holds Cl A meets Cl([#]GX \ A)
proof
A1: now
    given A being Subset of GX such that
A2: A <> {}GX and
A3: A <> [#]GX and
A4: (Cl A) misses Cl([#]GX \ A);
A5: (Cl A) /\ Cl([#]GX \ A) = {} by A4;
    A c= Cl A by PRE_TOPC:18;
    then A /\ Cl([#]GX \ A) = {}GX by A5,XBOOLE_1:3,27;
    then
A6: A misses Cl([#]GX \ A);
A7: [#]GX = A \/ (A`) by PRE_TOPC:2;
A8: [#]GX \ A <> {}GX by A3,PRE_TOPC:4;
    [#]GX \ A c= Cl([#]GX \ A) by PRE_TOPC:18;
    then (Cl A) /\ ([#]GX \ A) = {} by A5,XBOOLE_1:3,27;
    then (Cl A) misses ([#]GX \ A);
    then A,[#]GX \ A are_separated by A6;
    hence not GX is connected by A2,A8,A7;
  end;
  now
    assume not GX is connected;
    then consider A, B being Subset of GX such that
A9: [#]GX = A \/ B and
A10: A <> {}GX and
A11: B <> {}GX and
A12: A is closed and
A13: B is closed and
A14: A misses B by Th10;
A15: Cl A = A by A12,PRE_TOPC:22;
A16: Cl B = B by A13,PRE_TOPC:22;
A17: B = [#]GX \ A by A9,A14,PRE_TOPC:5;
    then A <> [#]GX by A11,PRE_TOPC:4;
    hence
    ex A being Subset of GX st A <> {}GX & A <> [#]GX & (Cl A) misses Cl(
    [#]GX \ A) by A10,A14,A17,A15,A16;
  end;
  hence thesis by A1;
end;
