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 y being ExtReal st
  y in Y ex x being ExtReal st x in X & x <= y holds inf X <= inf Y
proof
  let X,Y be ext-real-membered set;
  assume
A1: for y being ExtReal st y in Y ex x being ExtReal st
  x in X & x <= y;
  inf X is LowerBound of Y
  proof
    let y be ExtReal;
    assume y in Y;
    then consider x being ExtReal such that
A2: x in X and
A3: x <= y by A1;
    inf X <= x by A2,Th3;
    hence thesis by A3,XXREAL_0:2;
  end;
  hence thesis by Def4;
end;
