
theorem Th17:
  for T being non empty normal TopSpace, A,B being closed Subset
of T st A <> {} & A misses B holds for G being Rain of A,B, r1 being Element of
DOM st 0 < r1 holds for p being Point of T st r1 < (Thunder G).p holds not p in
  (Tempest G).r1
proof
  let T be non empty normal TopSpace;
  let A,B be closed Subset of T;
  assume
A1: A <> {} & A misses B;
  let G be Rain of A,B;
  let r1 be Element of DOM;
  assume
A2: 0 < r1;
  let p be Point of T;
  assume
A3: r1 < (Thunder G).p & p in (Tempest G).r1;
  r1 in halfline 0 \/ DYADIC or r1 in right_open_halfline 1 by URYSOHN1:def 3
,XBOOLE_0:def 3;
  then r1 in halfline 0 or r1 in DYADIC or r1 in right_open_halfline 1 by
XBOOLE_0:def 3;
  then r1 in DYADIC \/ right_open_halfline 1 by A2,XBOOLE_0:def 3,XXREAL_1:233;
  hence thesis by A1,A2,A3,Th16;
end;
