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

theorem
  for X being ext-real-membered set, x being ExtReal st (ex y
  being ExtReal st y in X & y <= x) holds inf X <= x
proof
  let X be ext-real-membered set;
  let x be ExtReal;
  given y being ExtReal such that
A1: y in X and
A2: y <= x;
  inf X <= y by A1,Th3;
  hence thesis by A2,XXREAL_0:2;
end;
