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

theorem Th25:
  for T being non empty 1-sorted, Y being net of T, J being
  net_set of the carrier of Y,T st Y in NetUniv T & for i being Element of Y
  holds J.i in NetUniv T holds Iterated J in NetUniv T
proof
  let T be non empty 1-sorted, Y be net of T, J be net_set of the carrier of Y
  ,T such that
A1: Y in NetUniv T and
A2: for i being Element of Y holds J.i in NetUniv T;
A3: rng Carrier J c= the_universe_of the carrier of T
  proof
    let x be object;
    assume x in rng Carrier J;
    then consider y being object such that
A4: y in dom Carrier J and
A5: (Carrier J).y = x by FUNCT_1:def 3;
    reconsider i = y as Element of Y by A4;
    J.i in NetUniv T by A2;
    then ex N being strict net of T st N = J.i & the carrier of N in
    the_universe_of the carrier of T by Def11;
    hence thesis by A5,Th2;
  end;
  the RelStr of Iterated J = [:Y, product J:] by Def13;
  then
A6: the carrier of Iterated J = [:the carrier of Y, the carrier of product J
  :] by YELLOW_3:def 2;
A7: ex N being strict net of T st N = Y & the carrier of N in
  the_universe_of the carrier of T by A1,Def11;
  then dom Carrier J in the_universe_of the carrier of T by PARTFUN1:def 2;
  then product Carrier J in the_universe_of the carrier of T by A3,Th1;
  then the carrier of product J in the_universe_of the carrier of T by
YELLOW_1:def 4;
  then
  the carrier of Iterated J in the_universe_of the carrier of T by A6,A7,
CLASSES2:61;
  hence thesis by Def11;
end;
