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
  C is connected & C meets A & C \ A <> {}GX implies C meets Fr A
proof
  assume that
A1: C is connected and
A2: C meets A and
A3: C \ A <> {}GX;
A4: A` c= Cl(A`) by PRE_TOPC:18;
  Cl (C /\ A) c= Cl A by PRE_TOPC:19,XBOOLE_1:17;
  then
A5: (Cl(C /\ A)) /\ (A`) c= (Cl A) /\ Cl(A`) by A4,XBOOLE_1:27;
A6: A c= Cl A by PRE_TOPC:18;
A7: C \ A = C /\ A` by SUBSET_1:13;
  then Cl (C \ A) c= Cl(A`) by PRE_TOPC:19,XBOOLE_1:17;
  then A /\ Cl(C /\ (A`)) c= (Cl A) /\ Cl(A`) by A7,A6,XBOOLE_1:27;
  then ((Cl(C /\ A)) /\ (A`)) \/ (A /\ Cl(C /\ (A`))) c= (Cl A) /\ Cl(A`) by A5
,XBOOLE_1:8;
  then
A8: C /\ (((Cl(C /\ A)) /\ (A`)) \/ (A /\ Cl(C /\ (A`)))) c= C /\ ((Cl A)
  /\ Cl(A`)) by XBOOLE_1:27;
A9: C = C /\ [#]GX by XBOOLE_1:28
    .= C /\ ( A \/ A`) by PRE_TOPC:2
    .= (C /\ A) \/ (C \ A) by A7,XBOOLE_1:23;
  C /\ A <> {} by A2;
  then not C /\ A,C \ A are_separated by A1,A3,A9,Th15;
  then (Cl(C /\ A)) meets (C \ A) or (C /\ A) meets Cl(C \ A);
  then
A10: (Cl(C /\ A)) /\ (C \ A) <> {} or (C /\ A) /\ Cl(C \ A) <> {};
  ((Cl(C /\ A)) /\ (C \ A)) \/ ((C /\ A) /\ Cl(C \ A)) = (((Cl(C /\ A))
  /\ C) /\ (A`)) \/ ((C /\ A) /\ Cl(C /\ (A`))) by A7,XBOOLE_1:16
    .= ((C /\ Cl(C /\ A)) /\ (A`)) \/ (C /\ (A /\ Cl(C /\ (A`)))) by
XBOOLE_1:16
    .= (C /\ ((Cl(C /\ A)) /\ (A`))) \/ (C /\ (A /\ Cl(C /\ (A`)))) by
XBOOLE_1:16
    .= C /\ ((Cl(C /\ A) /\ (A`)) \/ (A /\ Cl(C /\ A`))) by XBOOLE_1:23;
  then ((Cl(C /\ A)) /\ (C \ A)) \/ ((C /\ A) /\ Cl(C \ A)) c= C /\ Fr A by A8,
TOPS_1:def 2;
  hence C /\ Fr A <> {} by A10;
end;
