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

theorem Th11:
  for a, b, c being Real st a <= c & c <= b holds ]. -infty
  , c .] \/ [. a, b .] = ]. -infty, b .]
proof
  let a, b, c be Real;
  assume that
A1: a <= c and
A2: c <= b;
  thus ]. -infty, c .] \/ [. a, b .] c= ]. -infty, b .]
  proof
    let x be object;
    assume
A3: x in ]. -infty, c .] \/ [. a, b .];
    then reconsider x as Real;
    per cases by A3,XBOOLE_0:def 3;
    suppose
      x in ]. -infty, c .];
      then x <= c by XXREAL_1:234;
      then x <= b by A2,XXREAL_0:2;
      hence thesis by XXREAL_1:234;
    end;
    suppose
      x in [. a, b .];
      then x <= b by XXREAL_1:1;
      hence thesis by XXREAL_1:234;
    end;
  end;
  let x be object;
  assume
A4: x in ]. -infty, b .];
  then reconsider x as Real;
  per cases;
  suppose
    x <= c;
    then x in ]. -infty, c .] by XXREAL_1:234;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
A5: x > c;
A6: x <= b by A4,XXREAL_1:234;
    x > a by A1,A5,XXREAL_0:2;
    then x in [. a, b .] by A6,XXREAL_1:1;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
