
theorem Th40:
  for T being non empty RelStr, N being net of T for i being Element of N holds
  N|i is subnet of N & incl(N|i,N) is Embedding of N|i, N
proof
  let T be non empty RelStr, N be net of T;
  let i be Element of N;
  set M = N|i, f = incl(M,N);
  thus N|i is subnet of N;
  thus N|i is subnet of N;
  N|i is full SubNetStr of N by WAYBEL_9:14;
  then
A1: N|i is full SubRelStr of N by YELLOW_6:def 7;
A2: incl(N|i,N) = id the carrier of N|i by WAYBEL_9:13,YELLOW_9:def 1;
  the mapping of M = (the mapping of N)|the carrier of M by WAYBEL_9:def 7;
  hence the mapping of M = (the mapping of N)*f by A2,RELAT_1:65;
  let m be Element of N;
  consider n9 being Element of N such that
A3: n9 >= i and
A4: n9 >= m by YELLOW_6:def 3;
  reconsider n = n9 as Element of M by A3,WAYBEL_9:def 7;
  take n;
  let p be Element of M;
  reconsider p9 = p as Element of N by A1,YELLOW_0:58;
  assume n <= p;
  then n9 <= p9 by A1,YELLOW_0:59;
  then m <= p9 by A4,YELLOW_0:def 2;
  hence thesis by A2;
end;
