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 = ].
  -infty, a .] \/ {b} holds A` = ]. a, b .[ \/ ]. b,+infty .[
proof
  let A be Subset of R^1, a, b be Real;
  assume that
A1: a <= b and
A2: A = ]. -infty, a .] \/ {b};
  A` = (REAL \ ]. -infty, a .]) \ {b} by A2,TOPMETR:17,XBOOLE_1:41
    .= ]. a,+infty .[ \ {b} by XXREAL_1:224,288
    .= ]. a, b .[ \/ ]. b,+infty .[ by A1,XXREAL_1:365;
  hence thesis;
end;
