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

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