reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th57:
  for x,y being Real holds [.x,y.[ is closed Subset of Sorgenfrey-line
proof
  let x,y be Real;
  set T = Sorgenfrey-line;
  reconsider A = right_closed_halfline x, B = left_open_halfline y as closed
  Subset of T by Th54,Th56;
A1: the carrier of T = REAL by TOPGEN_3:def 2;
  [.x,y.[ = [.x,y.[``
    .= (left_open_halfline(x) \/ right_closed_halfline(y))` by XXREAL_1:382
    .= (left_open_halfline x)` /\ (right_closed_halfline y)` by XBOOLE_1:53
    .= A`` /\ (right_closed_halfline y)` by A1,XXREAL_1:224,294
    .= A /\ B by XXREAL_1:224,294;
  hence thesis;
end;
