
theorem Th16:
  for L being antisymmetric RelStr, X being set holds ex_inf_of X,
L iff ex a being Element of L st X is_>=_than a & for b being Element of L st X
  is_>=_than b holds a >= b
proof
  let L be antisymmetric RelStr, X be set;
  thus ex_inf_of X,L implies
   ex a being Element of L st X is_>=_than a & for b being Element of L
    st X is_>=_than b holds a >= b;
  given a being Element of L such that
A1: X is_>=_than a & for b being Element of L st X is_>=_than b holds a >= b;
  take a;
  thus X is_>=_than a & for b being Element of L st X is_>=_than b holds a >=
  b by A1;
  let c be Element of L;
  assume
  X is_>=_than c & for b being Element of L st X is_>=_than b holds c >= b;
  then a <= c & c <= a by A1;
  hence thesis by ORDERS_2:2;
end;
