
theorem
  for T being non empty RelStr
  for N being net of T, M being non empty full SubNetStr of N
  st for i being Element of N
  ex j being Element of N st j >= i & j in the carrier of M
  holds M is subnet of N & incl(M,N) is Embedding of M,N
proof
  let T be non empty RelStr;
  let N be net of T, M be non empty full SubNetStr of N such that
A1: for i being Element of N
  ex j being Element of N st j >= i & j in the carrier of M;
A2: the mapping of M = (the mapping of N)|the carrier of M by YELLOW_6:def 6;
A3: M is full SubRelStr of N by YELLOW_6:def 7;
  then
A4: the carrier of M c= the carrier of N by YELLOW_0:def 13;
  M is directed
  proof
    let x,y be Element of M;
    reconsider i = x, j = y as Element of N by A3,YELLOW_0:58;
    consider k being Element of N such that
A5: k >= i and
A6: k >= j by YELLOW_6:def 3;
    consider l being Element of N such that
A7: l >= k and
A8: l in the carrier of M by A1;
    reconsider z = l as Element of M by A8;
    take z;
A9: l >= i by A5,A7,YELLOW_0:def 2;
    l >= j by A6,A7,YELLOW_0:def 2;
    hence thesis by A9,YELLOW_6:12;
  end;
  then reconsider K = M as net of T by A3;
A10: now
    set f = incl(K,N);
A11: f = id the carrier of K by A4,YELLOW_9:def 1;
    hence the mapping of K = (the mapping of N)*f by A2,RELAT_1:65;
    let m be Element of N;
    consider j being Element of N such that
A12: j >= m and
A13: j in the carrier of K by A1;
    reconsider n = j as Element of K by A13;
    take n;
    let p be Element of K;
    reconsider q = p as Element of N by A3,YELLOW_0:58;
    assume n <= p;
    then
A14: j <= q by YELLOW_6:11;
    f.p = q by A11;
    hence m <= f.p by A12,A14,YELLOW_0:def 2;
  end;
  hence M is subnet of N by YELLOW_6:def 9;
  hence thesis by A10,Def3;
end;
