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

theorem
  for X,Y being ext-real-membered set holds sup(X /\ Y) <= min(sup X,sup Y)
proof
  let X,Y be ext-real-membered set;
A1: sup Y is UpperBound of Y by Def3;
  sup X is UpperBound of X by Def3;
  then min(sup X,sup Y) is UpperBound of X /\ Y by A1,Th65;
  hence thesis by Def3;
end;
