reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;
reserve A,B for ext-real-membered set;

theorem
  for A,B being non empty Subset of ExtREAL st for r,s being ExtReal
  st r in A & s in B holds r <= s holds sup A <= inf B
proof
  let A,B be non empty Subset of ExtREAL;
  assume
A1: for r,s being ExtReal st r in A & s in B holds r <= s;
  assume not sup A <= inf B;
  then consider s1 being Element of ExtREAL such that
A2: s1 in A and
A3: inf B < s1 by Th94;
  ex s2 being Element of ExtREAL st s2 in B & s2 < s1 by A3,Th95;
  hence contradiction by A1,A2;
end;
