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

theorem Th307:
  r < s & p <= q implies ].r,q.] \ ].p,s.] = ].r,p.] \/ ].s,q.]
proof
  assume that
A1: r < s and
A2: p <= q;
  let x be ExtReal;
  thus x in ].r,q.] \ ].p,s.] implies x in ].r,p.] \/ ].s,q.]
  proof
    assume
A3: x in ].r,q.] \ ].p,s.];
    then
A4: not x in ].p,s.] by XBOOLE_0:def 5;
A5: r < x by A3,Th2;
A6: x <= q by A3,Th2;
    not(p < x & x <= s) by A4,Th2;
    then x in ].r,p.] or x in ].s,q.] by A5,A6,Th2;
    hence thesis by XBOOLE_0:def 3;
  end;
  assume x in ].r,p.] \/ ].s,q.];
  then x in ].r,p.] or x in ].s,q.] by XBOOLE_0:def 3;
  then
A7: r < x & x <= p or s < x & x <= q by Th2;
  then
A8: r < x by A1,XXREAL_0:2;
  x <= q by A2,A7,XXREAL_0:2;
  then
A9: x in ].r,q.] by A8,Th2;
  not x in ].p,s.] by A7,Th2;
  hence thesis by A9,XBOOLE_0:def 5;
end;
