
theorem Th15:
  for L being antisymmetric RelStr, X being set holds ex_sup_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_sup_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 b >=
  a 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 b >= c;
  then a <= c & c <= a by A1;
  hence thesis by ORDERS_2:2;
end;
