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 implies for S being Subset of GX st S is a_component holds
  C misses S or C c= S
proof
  assume
A1: C is connected;
  let S be Subset of GX;
  assume
A2: S is a_component;
A3: S c= C \/ S by XBOOLE_1:7;
  assume C meets S;
  then C \/ S is connected by A1,A2,Th1,Th17;
  then S = C \/ S by A2,A3;
  hence thesis by XBOOLE_1:7;
end;
