
theorem
  for L being non empty 1-sorted, N,M being net of L st the NetStr of N
  = the NetStr of M holds M is subnet of N
proof
  let L be non empty 1-sorted, N,M be net of L;
  assume
A1: the NetStr of N = the NetStr of M;
A2: N is subnet of N by YELLOW_6:14;
  ex f being Function of M, N st the mapping of M = (the mapping of N)*f &
for m being Element of N ex n being Element of M st for p being Element of M st
  n <= p holds m <= f.p
  proof
    consider f being Function of N, N such that
A3: the mapping of N = (the mapping of N)*f and
A4: for m being Element of N ex n being Element of N st for p being
    Element of N st n <= p holds m <= f.p by A2,YELLOW_6:def 9;
    reconsider f2=f as Function of M,N by A1;
    take f2;
    thus the mapping of M = (the mapping of N)*f2 by A1,A3;
    let m be Element of N;
    consider n being Element of N such that
A5: for p being Element of N st n <= p holds m <= f.p by A4;
    reconsider n2=n as Element of M by A1;
    take n2;
    let p be Element of M;
    reconsider p1=p as Element of N by A1;
    assume n2 <= p;
    then [n2,p] in the InternalRel of M by ORDERS_2:def 5;
    then n <= p1 by A1,ORDERS_2:def 5;
    hence thesis by A5;
  end;
  hence thesis by YELLOW_6:def 9;
end;
