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

theorem
  r <= s implies [.r,s.] <> [.p,q.[
proof
  assume that
A1: r <= s and
A2: [.r,s.] = [.p,q.[;
  now
    assume s in [.p,q.[;
    then
A3: s < q by Th3;
    p <= r by A1,A2,Th54;
    then p <= s by A1,XXREAL_0:2;
    then p < q by A3,XXREAL_0:2;
    hence contradiction by A2,A3,Th52;
  end;
  hence contradiction by A1,A2,Th1;
end;
