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,Th1;
A5: t <= s by A2,Th1;
A6: p <= t by A3,Th1;
A7: t <= q by A3,Th1;
A8: max(r,p) <= t by A4,A6,XXREAL_0:28;
    t <= min(s,q) by A5,A7,XXREAL_0:20;
    hence thesis by A8,Th1;
  end;
  assume
A9: t in [.max(r,p),min(s,q).];
  then
A10: max(r,p) <= t by Th1;
A11: t <= min(s,q) by A9,Th1;
A12: r <= t by A10,XXREAL_0:30;
A13: p <= t by A10,XXREAL_0:30;
A14: t <= s by A11,XXREAL_0:22;
A15: t <= q by A11,XXREAL_0:22;
A16: t in [.r,s.] by A12,A14,Th1;
  t in [.p,q.] by A13,A15,Th1;
  hence thesis by A16,XBOOLE_0:def 4;
end;
