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

theorem Th43:
  for a being Real holds ]. -infty,a.] = {a} \/ ]. -infty,a .[
proof
  let a be Real;
  thus ]. -infty,a.] c= {a} \/ ]. -infty,a.[
  proof
    let x be object;
    assume
A1: x in ]. -infty,a.];
    then reconsider x as Real;
A2: x <= a by A1,XXREAL_1:234;
    per cases by A2,XXREAL_0:1;
    suppose
      x = a;
      then x in {a} by TARSKI:def 1;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      x < a;
      then x in ]. -infty,a.[ by XXREAL_1:233;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  let x be object;
  assume
A3: x in {a} \/ ]. -infty,a.[;
  then reconsider x as Real;
  x in {a} or x in ]. -infty,a.[ by A3,XBOOLE_0:def 3;
  then x = a or x < a by TARSKI:def 1,XXREAL_1:233;
  hence thesis by XXREAL_1:234;
end;
