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

theorem
  for a, b being Real holds ]. -infty, a .] \/ {b} <> REAL
proof
  let a, b be Real;
  set ab = max (a,b) + 1;
A1: ab > max (a,b) by XREAL_1:29;
  max (a,b) >= a by XXREAL_0:25;
  then ab > a by A1,XXREAL_0:2;
  then
A2: not ab in ]. -infty, a.] by XXREAL_1:234;
  max (a,b) >= b by XXREAL_0:25;
  then
A3: not ab in {b} by A1,TARSKI:def 1;
  ab in REAL by XREAL_0:def 1;
  hence thesis by A2,A3,XBOOLE_0:def 3;
end;
