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

theorem Th10:
  for a, b, c being Real st a <= c & c <= b holds [.a,b.]
  \/ [.c,+infty .[ = [.a,+infty .[
proof
  let a, b, c be Real;
  assume that
A1: a <= c and
A2: c <= b;
A3: [.a,+infty .[ c= [.a,b.] \/ [.c,+infty .[
  proof
    let r be object;
    assume
A4: r in [.a,+infty .[;
    then reconsider s = r as Real;
A5: a <= s by A4,XXREAL_1:236;
    per cases;
    suppose
      s <= b;
      then s in [.a,b.] by A5,XXREAL_1:1;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      not s <= b;
      then c <= s by A2,XXREAL_0:2;
      then s in [.c,+infty .[ by XXREAL_1:236;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
A6: [.a,b.] c= right_closed_halfline a by XXREAL_1:251;
  [.c,+infty .[ c= [.a,+infty .[ by A1,XXREAL_1:38;
  then [.a,b.] \/ [.c,+infty .[ c= [.a,+infty .[ by A6,XBOOLE_1:8;
  hence thesis by A3,XBOOLE_0:def 10;
end;
