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 Th18:
  C is connected & C c= A & A c= Cl C implies A is connected
proof
  assume that
A1: C is connected and
A2: C c= A and
A3: A c= Cl C;
  assume not A is connected;
  then consider P,Q being Subset of GX such that
A4: A = P \/ Q and
A5: P,Q are_separated and
A6: P <> {}GX and
A7: Q <> {}GX by Th15;
  P misses Cl Q by A5;
  then
A8: P /\ Cl Q = {};
A9: now
    assume C c= Q;
    then Cl C c= Cl Q by PRE_TOPC:19;
    then P /\ Cl C = {} by A8,XBOOLE_1:3,27;
    then P /\ A = {} by A3,XBOOLE_1:3,27;
    hence contradiction by A4,A6,XBOOLE_1:21;
  end;
  (Cl P) misses Q by A5;
  then
A10: (Cl P) /\ Q = {};
  now
    assume C c= P;
    then Cl C c= Cl P by PRE_TOPC:19;
    then (Cl C) /\ Q = {} by A10,XBOOLE_1:3,27;
    then A /\ Q = {} by A3,XBOOLE_1:3,27;
    hence contradiction by A4,A7,XBOOLE_1:21;
  end;
  hence contradiction by A1,A2,A4,A5,A9,Th16;
end;
