
theorem
  for L being non empty Poset, D being non empty finite filtered Subset
  of L st ex_inf_of D,L holds inf D in D
proof
  let L be non empty Poset, D be non empty finite filtered Subset of L such
  that
A1: ex_inf_of D,L;
  D c= D;
  then consider d being Element of L such that
A2: d in D and
A3: d is_<=_than D by WAYBEL_0:2;
A4: inf D >= d by A1,A3,YELLOW_0:31;
  inf D is_<=_than D by A1,YELLOW_0:31;
  then inf D <= d by A2;
  hence thesis by A2,A4,ORDERS_2:2;
end;
