reserve GX,GY for non empty TopSpace,
  x,y for Point of GX,
  r,s for Real;

theorem Th5:
  for A0,A1,A2,A3 being Subset of GX st A0 is connected & A1 is connected &
  A2 is connected & A3 is connected & A0 meets A1 & A1 meets A2 &
  A2 meets A3 holds A0 \/ A1 \/ A2 \/ A3 is connected
proof
  let A0,A1,A2,A3 be Subset of GX;
  assume that
A1: A0 is connected and
A2: A1 is connected and
A3: A2 is connected and
A4: A3 is connected and
A5: A0 meets A1 and
A6: A1 meets A2 and
A7: A2 meets A3;
A8: A2 /\ A3 <> {} by A7;
A9: A0 \/ A1 \/ A2 is connected by A1,A2,A3,A5,A6,Th4;
  (A0 \/ A1 \/ A2)/\ A3= (A0 \/ A1) /\ A3 \/ A2 /\ A3 by XBOOLE_1:23;
  then (A0 \/ A1 \/ A2) meets A3 by A8;
  hence thesis by A4,A9,CONNSP_1:1,17;
end;
