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

theorem
  ].r,s.] /\ ].p,q.] = ].max(r,p),min(s,q).]
proof
  let t;
  thus t in ].r,s.] /\ ].p,q.] implies t in ].max(r,p),min(s,q).]
  proof
    assume
A1: t in ].r,s.] /\ ].p,q.];
    then
A2: t in ].r,s.] by XBOOLE_0:def 4;
A3: t in ].p,q.] by A1,XBOOLE_0:def 4;
A4: r < t by A2,Th2;
A5: t <= s by A2,Th2;
A6: p < t by A3,Th2;
A7: t <= q by A3,Th2;
A8: max(r,p) < t by A4,A6,XXREAL_0:29;
    t <= min(s,q) by A5,A7,XXREAL_0:20;
    hence thesis by A8,Th2;
  end;
  assume
A9: t in ].max(r,p),min(s,q).];
  then
A10: max(r,p) < t by Th2;
A11: t <= min(s,q) by A9,Th2;
A12: r < t by A10,XXREAL_0:31;
A13: p < t by A10,XXREAL_0:31;
A14: t <= s by A11,XXREAL_0:22;
A15: t <= q by A11,XXREAL_0:22;
A16: t in ].r,s.] by A12,A14,Th2;
  t in ].p,q.] by A13,A15,Th2;
  hence thesis by A16,XBOOLE_0:def 4;
end;
