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

theorem Th42:
  for a being Real holds [. a,+infty .[ = {a} \/ ]. a, +infty .[
proof
  let a be Real;
  thus [. a,+infty .[ c= {a} \/ ]. a,+infty .[
  proof
    let x be object;
    assume
A1: x in [. a,+infty .[;
    then reconsider x as Real;
A2: x >= a by A1,XXREAL_1:236;
    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 ]. a,+infty .[ by XXREAL_1:235;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  let x be object;
  assume
A3: x in {a} \/ ]. a,+infty .[;
  then reconsider x as Real;
  x in {a} or x in ]. a,+infty .[ by A3,XBOOLE_0:def 3;
  then x = a or x > a by TARSKI:def 1,XXREAL_1:235;
  hence thesis by XXREAL_1:236;
end;
