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

theorem
  for A being Subset of R^1, a, b being Real st a <= b & A = {a}
  \/ [. b,+infty .[ holds A` = ]. -infty, a .[ \/ ]. a, b .[
proof
  let A be Subset of R^1, a, b be Real;
  assume that
A1: a <= b and
A2: A = {a} \/ [. b,+infty .[;
  A` = (REAL \ [. b,+infty .[) \ {a} by A2,TOPMETR:17,XBOOLE_1:41
    .= ]. -infty,b.[ \ {a} by XXREAL_1:224,294;
  hence thesis by A1,XXREAL_1:349;
end;
