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

theorem
  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 Th2,Th3;
    then
A4: r < p by A1,XXREAL_0:2;
    p < t by A2,A3,XXREAL_0:2;
    hence thesis by A4,Th4;
  end;
  assume p in ].r,t.[;
  then r < p & p <= s or s < p & p < t by Th4;
  then p in ].r,s.] or p in [.s,t.[ by Th2,Th3;
  hence thesis by XBOOLE_0:def 3;
end;
