
theorem Th17:
  for L being non empty 1-sorted, N being net of L, i being
  Element of N holds N|i is subnet of N
proof
  let L be non empty 1-sorted, N be net of L, i be Element of N;
  reconsider A = N|i as net of L;
A1: the carrier of A c= the carrier of N by Th13;
  A is subnet of N
  proof
    reconsider f = id the carrier of A as Function of A, N by A1,FUNCT_2:7;
    take f;
    for x being object st x in the carrier of A holds (the mapping of A).x =
    ((the mapping of N)*f).x
    proof
      let x be object such that
A2:   x in the carrier of A;
      thus (the mapping of A).x = ((the mapping of N)|the carrier of A).x by
Def7
        .= (the mapping of N).x by A2,FUNCT_1:49
        .= (the mapping of N).(f.x) by A2,FUNCT_1:18
        .= ((the mapping of N)*f).x by A2,FUNCT_2:15;
    end;
    hence the mapping of A = (the mapping of N)*f by FUNCT_2:12;
    let m be Element of N;
    consider z being Element of N such that
A3: i <= z and
A4: m <= z by YELLOW_6:def 3;
    reconsider n = z as Element of A by A3,Def7;
    take n;
    let p be Element of A such that
A5: n <= p;
    reconsider p1 = p as Element of N by A1;
    A is SubNetStr of N by Th14;
    then z <= p1 by A5,YELLOW_6:11;
    then m <= p1 by A4,YELLOW_0:def 2;
    hence thesis;
  end;
  hence thesis;
end;
