reserve x for set,
  p,q,r,s,t,u for ExtReal,
  g for Real,
  a for Element of ExtREAL;

theorem Th166:
  r <= s & s <= t implies [.r,s.[ \/ [.s,t.] = [.r,t.]
proof
  assume that
A1: r <= s and
A2: s <= t;
  let p;
  thus p in [.r,s.[ \/ [.s,t.] implies p in [.r,t.]
  proof
    assume p in [.r,s.[ \/ [.s,t.];
    then p in [.r,s.[ or p in [.s,t.] by XBOOLE_0:def 3;
    then
A3: r <= p & p < s or s <= p & p <= t by Th1,Th3;
    then
A4: r <= p by A1,XXREAL_0:2;
    p <= t by A2,A3,XXREAL_0:2;
    hence thesis by A4,Th1;
  end;
  assume p in [.r,t.];
  then r <= p & p < s or s <= p & p <= t by Th1;
  then p in [.r,s.[ or p in [.s,t.] by Th1,Th3;
  hence thesis by XBOOLE_0:def 3;
end;
