
theorem
  for X,Y being ext-real-membered set st X c= Y holds for x being
  ExtReal holds x in SetMinorant(Y) implies x in SetMinorant(X)
proof
  let X,Y be ext-real-membered set;
  assume
A1: X c= Y;
  let x be ExtReal;
  assume x in SetMinorant(Y);
  then x is LowerBound of Y by Def2;
  then x is LowerBound of X by A1,XXREAL_2:5;
  hence thesis by Def2;
end;
