reserve x for set;
reserve a,b,d,ra,rb,r0,s1,s2 for Real;
reserve r,s,r1,r2,r3,rc for Real;
reserve p,q,q1,q2 for Point of TOP-REAL 2;
reserve X,Y,Z for non empty TopSpace;

theorem Th1:
  for A,B1,B2 being Subset of X st B1 is open & B2 is open & B1
  meets A & B2 meets A & A c= B1 \/ B2 & B1 misses B2 holds A is not connected
proof
  let A,B1,B2 be Subset of X;
  assume that
A1: B1 is open & B2 is open & B1 meets A and
A2: B2 meets A and
A3: A c= B1 \/ B2 and
A4: B1 misses B2;
  reconsider C1=B1 /\ A, C2=B2 /\ A as Subset of X;
A5: A=(B1 \/ B2)/\A by A3,XBOOLE_1:28
    .=C1 \/ C2 by XBOOLE_1:23;
A6: C2 <> {} by A2,XBOOLE_0:def 7;
A7: A is connected iff for P,Q being Subset of X st A = P \/ Q & P,Q
  are_separated holds P = {}X or Q = {}X by CONNSP_1:15;
A8: C1 c= B1 & C2 c= B2 by XBOOLE_1:17;
  B1,B2 are_separated & C1 <> {} by A1,A4,TSEP_1:37,XBOOLE_0:def 7;
  hence thesis by A7,A5,A8,A6,CONNSP_1:7;
end;
