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

theorem
  for Q being 1-sorted, R being NetStr over Q, S being SubNetStr of R, T
  being SubNetStr of S holds T is SubNetStr of R
proof
  let Q be 1-sorted, R be NetStr over Q, S be SubNetStr of R, T be SubNetStr
  of S;
A1: T is SubRelStr of S by Def6;
  then
A2: the carrier of T c= the carrier of S by YELLOW_0:def 13;
A3: the mapping of T = (the mapping of S)|the carrier of T by Def6
    .= (the mapping of R)|(the carrier of S)|the carrier of T by Def6
    .= (the mapping of R)|((the carrier of S) /\ the carrier of T) by
RELAT_1:71
    .= (the mapping of R)|the carrier of T by A2,XBOOLE_1:28;
  S is SubRelStr of R by Def6;
  then T is SubRelStr of R by A1,Th7;
  hence thesis by A3,Def6;
end;
