
theorem Th27:
  for L being with_infima antisymmetric transitive RelStr, N being
prenet of L for x being Element of L st N is eventually-directed holds x "/\" N
  is eventually-directed
proof
  let L be with_infima antisymmetric transitive RelStr, N be prenet of L, x be
  Element of L such that
A1: N is eventually-directed;
A2: the RelStr of x "/\" N = the RelStr of N by Def3;
  for i being Element of x "/\" N ex j being Element of x "/\" N st for k
  being Element of x "/\" N st j <= k holds (x "/\" N).i <= (x "/\" N).k
  proof
    let i1 be Element of x "/\" N;
    reconsider i = i1 as Element of N by Th22;
    consider j being Element of N such that
A3: for k being Element of N st j <= k holds N.i <= N.k by A1,WAYBEL_0:11;
    reconsider j1 = j as Element of x "/\" N by Th22;
    take j1;
    let k1 be Element of x "/\" N;
    reconsider k = k1 as Element of N by Th22;
    assume j1 <= k1;
    then j <= k by A2,YELLOW_0:1;
    then
A4: N.i <= N.k by A3;
    ( ex yi being Element of L st yi = (the mapping of N).i1 & (the
    mapping of x "/\" N).i1 = x "/\" yi)& ex yk being Element of L st yk = (the
    mapping of N). k1 & (the mapping of x "/\" N).k1 = x "/\" yk by Def3;
    hence thesis by A4,WAYBEL_1:1;
  end;
  hence thesis by WAYBEL_0:11;
end;
