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

theorem
  q < s & r <= s implies not [.r,s.] c= [.p,q.]
proof
  assume that
A1: q < s and
A2: r <= s;
  per cases;
  suppose
A3: r <= q;
    consider t such that
A4: q < t and
A5: t < s by A1,XREAL_1:227;
    take t;
    r < t by A3,A4,XXREAL_0:2;
    hence t in [.r,s.] by A5,Th1;
    thus thesis by A4,Th1;
  end;
  suppose
A6: q < r;
    take t=r;
    thus t in [.r,s.] by A2,Th1;
    thus thesis by A6,Th1;
  end;
end;
