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 st for x being ExtReal st
  x in X ex y being ExtReal st y in Y & x <= y holds sup X <= sup Y
proof
  let X,Y be ext-real-membered set;
  assume
A1: for x being ExtReal st x in X ex y being ExtReal st
  y in Y & x <= y;
  sup Y is UpperBound of X
  proof
    let x be ExtReal;
    assume x in X;
    then consider y being ExtReal such that
A2: y in Y and
A3: x <= y by A1;
    y <= sup Y by A2,Th4;
    hence thesis by A3,XXREAL_0:2;
  end;
  hence thesis by Def3;
end;
