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

theorem
  r <= p & s <= q implies [.r,s.[ /\ [.p,q.[ = [.p,s.[
proof
  assume that
A1: r <= p and
A2: s <= q;
  let t;
  thus t in [.r,s.[ /\ [.p,q.[ implies t in [.p,s.[
  proof
    assume
A3: t in [.r,s.[ /\ [.p,q.[;
    then
A4: t in [.r,s.[ by XBOOLE_0:def 4;
A5: t in [.p,q.[ by A3,XBOOLE_0:def 4;
A6: t < s by A4,Th3;
    p <= t by A5,Th3;
    hence thesis by A6,Th3;
  end;
  assume
A7: t in [.p,s.[;
  then
A8: p <= t by Th3;
A9: t < s by A7,Th3;
A10: r <= t by A1,A8,XXREAL_0:2;
A11: t < q by A2,A9,XXREAL_0:2;
A12: t in [.r,s.[ by A9,A10,Th3;
  t in [.p,q.[ by A8,A11,Th3;
  hence thesis by A12,XBOOLE_0:def 4;
end;
