reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th27: ::3.12 (1-4), p.70
  for L being distributive LATTICE for l being Element of L holds
  l is prime iff l is irreducible
proof
  let L be distributive LATTICE,l be Element of L;
  thus l is prime implies l is irreducible by Th24;
  thus l is irreducible implies l is prime
  proof
    assume
A1: l is irreducible;
    let x,y be Element of L;
    assume x "/\" y <= l;
    then l = l "\/" (x "/\" y) by YELLOW_0:24
      .= (l "\/" x) "/\" (l "\/" y) by WAYBEL_1:5;
    then l = l "\/" x or l = l "\/" y by A1;
    hence x <= l or y <= l by YELLOW_0:24;
  end;
end;
