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

theorem Th178:
  p < s & r < q & s <= r implies ].p,r.] \/ [.s,q.[ = ].p,q.[
proof
  assume that
A1: p < s and
A2: r < q and
A3: s <= r;
  let t;
  thus t in ].p,r.] \/ [.s,q.[ implies t in ].p,q.[
  proof
    assume t in ].p,r.] \/ [.s,q.[;
    then t in ].p,r.] or t in [.s,q.[ by XBOOLE_0:def 3;
    then
A4: p < t & t <= r or s <= t & t < q by Th2,Th3;
    then
A5: p < t by A1,XXREAL_0:2;
    t < q by A2,A4,XXREAL_0:2;
    hence thesis by A5,Th4;
  end;
  assume t in ].p,q.[;
  then p < t & t <= r or s <= t & t < q by A3,Th4,XXREAL_0:2;
  then t in ].p,r.] or t in [.s,q.[ by Th2,Th3;
  hence thesis by XBOOLE_0:def 3;
end;
