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
  GX is connected & A is connected & C is_a_component_of [#]GX \ A
  implies [#]GX \ C is connected
proof
  assume that
A1: GX is connected and
A2: A is connected and
A3: C is_a_component_of [#]GX \ A;
  consider C1 being Subset of GX|([#]GX \ A) such that
A4: C1 = C and
A5: C1 is a_component by A3;
  reconsider C2 = C1 as Subset of GX by A4;
  C1 c= [#](GX|([#]GX \ A));
  then C1 c= [#]GX \ A by PRE_TOPC:def 5;
  then ([#]GX \ A)` c= C2` by SUBSET_1:12;
  then
A6: A c= C2` by PRE_TOPC:3;
  now
    A misses C1 by A6,SUBSET_1:23;
    then
A7: A /\ C1 = {};
A8: C is connected by A4,A5,Th23;
    let P,Q be Subset of GX such that
A9: [#]GX \ C = P \/ Q and
A10: P,Q are_separated;
A11: P misses P` by XBOOLE_1:79;
A12: P misses Q by A10,Th1;
A13: now
A14:  Q misses Q` by XBOOLE_1:79;
      assume
A15:  A c= Q;
      P c= Q` by A12,SUBSET_1:23;
      then A /\ P c= Q /\ Q` by A15,XBOOLE_1:27;
      then
A16:  A /\ P c= {} by A14;
      (C \/ P ) /\ A = (A /\ C) \/ (A /\ P) by XBOOLE_1:23
        .= {} by A4,A7,A16;
      then C \/ P misses A;
      then C \/ P c= A` by SUBSET_1:23;
      then C \/ P c= [#](GX|([#]GX \ A)) by PRE_TOPC:def 5;
      then reconsider C1P1 = C \/ P as Subset of GX|([#]GX \ A);
A17:  C misses C` by XBOOLE_1:79;
      C \/ P is connected by A1,A9,A10,A8,Th20;
      then
A18:  C1P1 is connected by Th23;
      C c= C1 \/ P by A4,XBOOLE_1:7;
      then C1P1 = C1 by A4,A5,A18;
      then
A19:  P c= C by A4,XBOOLE_1:7;
      P c= [#]GX \ C by A9,XBOOLE_1:7;
      then P c= C /\ ([#]GX \ C) by A19,XBOOLE_1:19;
      then P c= {} by A17;
      hence P = {}GX;
    end;
A20: Q c= [#]GX \ C by A9,XBOOLE_1:7;
    now
      assume
A21:  A c= P;
      Q c= P` by A12,SUBSET_1:23;
      then A /\ Q c= P /\ P` by A21,XBOOLE_1:27;
      then
A22:  A /\ Q c= {} by A11;
      (C \/ Q) /\ A = (A /\ C) \/ (A /\ Q) by XBOOLE_1:23
        .= {} by A4,A7,A22;
      then (C \/ Q) misses A;
      then C \/ Q c= A` by SUBSET_1:23;
      then C \/ Q c= [#](GX|([#]GX \ A)) by PRE_TOPC:def 5;
      then reconsider C1Q1 = C \/ Q as Subset of GX|([#]GX \ A);
      C \/ Q is connected by A1,A9,A10,A8,Th20;
      then
A23:  C1Q1 is connected by Th23;
      C1 c= C1 \/ Q by XBOOLE_1:7;
      then C1Q1 = C1 by A4,A5,A23;
      then Q c= C by A4,XBOOLE_1:7;
      then
A24:  Q c= C /\ ([#]GX \ C) by A20,XBOOLE_1:19;
      C misses C` by XBOOLE_1:79;
      then Q c= {} by A24;
      hence Q = {}GX;
    end;
    hence P = {}GX or Q = {}GX by A2,A4,A6,A9,A10,A13,Th16;
  end;
  hence thesis by Th15;
end;
