reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th52:
  for A, B being Subset of R^1, b being Real st A = ].
  -infty, b .[ & B = ]. b,+infty .[ holds A, B are_separated
proof
  let A, B be Subset of R^1, b be Real;
  assume that
A1: A = ]. -infty, b .[ and
A2: B = ]. b,+infty .[;
  Cl B = [. b,+infty .[ by A2,Th48;
  then
A3: Cl B misses A by A1,XXREAL_1:94;
  Cl A = ]. -infty, b .] by A1,Th50;
  then Cl A misses B by A2,XXREAL_1:91;
  hence thesis by A3,CONNSP_1:def 1;
end;
