
theorem Th17:
  for L being complete non empty antisymmetric RelStr, X being set
  holds ex_sup_of X,L & ex_inf_of X,L
proof
  let L be complete non empty antisymmetric RelStr, X be set;
  set Y = {c where c is Element of L: c is_<=_than X};
  consider a being Element of L such that
A1: Y is_<=_than a and
A2: for b being Element of L st Y is_<=_than b holds a <= b by LATTICE3:def 12;
  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 by LATTICE3:def 12;
  hence ex_sup_of X,L by Th15;
  now
    take a;
    thus a is_<=_than X
    proof
      let b be Element of L;
      assume
A3:   b in X;
      Y is_<=_than b
      proof
        let c be Element of L;
        assume c in Y;
        then ex d being Element of L st c = d & d is_<=_than X;
        hence thesis by A3;
      end;
      hence thesis by A2;
    end;
    let b be Element of L;
    assume b is_<=_than X;
    then b in Y;
    hence b <= a by A1;
  end;
  hence thesis by Th16;
end;
