reserve x,y,z,X for set,
  T for Universe;

theorem
  for T being non empty 1-sorted, N1,N2,N3 be net of T st N1 is subnet
  of N2 & N2 is subnet of N3 holds N1 is subnet of N3
proof
  let T be non empty 1-sorted, N1,N2,N3 be net of T;
  given f being Function of N1, N2 such that
A1: the mapping of N1 = (the mapping of N2)*f and
A2: for m being Element of N2 ex n being Element of N1 st for p being
  Element of N1 st n <= p holds m <= f.p;
  given g being Function of N2, N3 such that
A3: the mapping of N2 = (the mapping of N3)*g and
A4: for m being Element of N3 ex n being Element of N2 st for p being
  Element of N2 st n <= p holds m <= g.p;
  take g*f;
  thus the mapping of N1 = (the mapping of N3)*(g*f) by A1,A3,RELAT_1:36;
  let m be Element of N3;
  consider m1 being Element of N2 such that
A5: for p being Element of N2 st m1 <= p holds m <= g.p by A4;
  consider n being Element of N1 such that
A6: for p being Element of N1 st n <= p holds m1 <= f.p by A2;
  take n;
  let p be Element of N1;
  assume n <= p;
  then m <= g.(f.p) by A5,A6;
  hence thesis by FUNCT_2:15;
end;
