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

theorem
  ].p,q.] meets ].r,s.] implies ].p,q.] \ ].r,s.] = ].p,r.] \/ ].s,q.]
proof
  assume ].p,q.] meets ].r,s.];
  then consider u such that
A1: u in ].r,s.] and
A2: u in ].p,q.];
A3: r < u by A1,Th2;
A4: u <= s by A1,Th2;
A5: p < u by A2,Th2;
  u <= q by A2,Th2;
  then
A6: r <= q by A3,XXREAL_0:2;
A7: p <= s by A4,A5,XXREAL_0:2;
  let t;
  thus t in ].p,q.] \ ].r,s.] implies t in ].p,r.] \/ ].s,q.]
  proof
    assume
A8: t in ].p,q.] \ ].r,s.];
    then
A9: not t in ].r,s.] by XBOOLE_0:def 5;
A10: p < t by A8,Th2;
A11: t <= q by A8,Th2;
    t <= r or s < t by A9,Th2;
    then t in ].p,r.] or t in ].s,q.] by A10,A11,Th2;
    hence thesis by XBOOLE_0:def 3;
  end;
  assume t in ].p,r.] \/ ].s,q.];
  then t in ].p,r.] or t in ].s,q.] by XBOOLE_0:def 3;
  then
A12: p < t & t <= r or s < t & t <= q by Th2;
  then
A13: p < t by A7,XXREAL_0:2;
  t <= q by A6,A12,XXREAL_0:2;
  then
A14: t in ].p,q.] by A13,Th2;
  not t in ].r,s.] by A12,Th2;
  hence thesis by A14,XBOOLE_0:def 5;
end;
