
theorem Th10:
  for L being non empty RelStr, a being Element of L, N being net
  of L holds a "/\" N is net of L
proof
  let L be non empty RelStr, a be Element of L, N be net of L;
  set aN = a "/\" N;
  aN is transitive
  proof
    let x, y, z be Element of aN such that
A1: x <= y & y <= z;
    reconsider x1 = x, y1 = y, z1 = z as Element of N by WAYBEL_2:22;
A2: the RelStr of N = the RelStr of aN by WAYBEL_2:def 3;
    then x1 <= y1 & y1 <= z1 by A1;
    then x1 <= z1 by YELLOW_0:def 2;
    hence thesis by A2;
  end;
  hence thesis;
end;
