
theorem Th12:
  for L being non empty 1-sorted, N being net of L, p being
  greater_or_equal_to_id Function of N,N holds N * p is subnet of N
proof
  let L be non empty 1-sorted;
  let N be net of L;
  let p be greater_or_equal_to_id Function of N,N;
  ex f being Function of (N * p), N st the mapping of (N * p) = (the
mapping of N)*f & for m being Element of N ex n being Element of (N * p) st for
  k being Element of (N * p) st n <= k holds m <= f.k
  proof
    the carrier of N * p = the carrier of N by Th6;
    then reconsider f=p as Function of (N * p), N;
    take f;
    thus the mapping of (N * p) = (the mapping of N)*f by Def2;
    let m be Element of N;
    reconsider n=m as Element of (N * p) by Th6;
    take n;
    let k be Element of (N * p);
    assume
A1: n <= k;
    reconsider k1 = k as Element of N by Th6;
A2: k1 <= p.k1 by Def1;
    the RelStr of N*p = the RelStr of N by Def2;
    then m <= k1 by A1,YELLOW_0:1;
    hence thesis by A2,YELLOW_0:def 2;
  end;
  hence thesis by YELLOW_6:def 9;
end;
