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

theorem Th28:
  for T being non empty 1-sorted, Y being net of T, J being
  net_set of the carrier of Y,T holds rng the mapping of Iterated J c= union
the set of all
  rng the mapping of J.i where i is Element of Y
proof
  let T be non empty 1-sorted, Y be net of T, J be net_set of the carrier of Y
  ,T;
  let x be object;
  set X = the set of all  rng the mapping of J.i where i is Element of Y;
  assume x in rng the mapping of Iterated J;
  then consider y being object such that
A1: y in dom the mapping of Iterated J and
A2: (the mapping of Iterated J).y = x by FUNCT_1:def 3;
  y in the carrier of Iterated J by A1;
  then y in [:the carrier of Y, product Carrier J:] by Th26;
  then consider
  y1 being Element of Y, y2 being Element of product Carrier J such
  that
A3: y = [y1,y2] by DOMAIN_1:1;
  y1 in the carrier of Y;
  then y1 in dom Carrier J by PARTFUN1:def 2;
  then y2.y1 in (Carrier J).y1 by CARD_3:9;
  then y2.y1 in the carrier of J.y1 by Th2;
  then
A4: y2.y1 in dom the mapping of J.y1 by FUNCT_2:def 1;
  y2 in product Carrier J;
  then
A5: y2 in the carrier of product J by YELLOW_1:def 4;
  x = (the mapping of Iterated J).(y1,y2) by A2,A3
    .= (the mapping of J.y1).(y2.y1) by A5,Def13;
  then
A6: x in rng the mapping of J.y1 by A4,FUNCT_1:def 3;
  reconsider y1 as Element of Y;
  rng the mapping of J.y1 in X;
  hence thesis by A6,TARSKI:def 4;
end;
