theorem Th72:
  for s being positive Nat
  for f being Solution_of_Sierp168 of s holds
  rng Solutions_of_Sierp168(f) c=
  the set of all g where g is Solution_of_Sierp168 of s
  proof
    let s be positive Nat;
    let f be Solution_of_Sierp168 of s;
    set F = Solutions_of_Sierp168(f);
    let y be object;
    assume y in rng F;
    then consider x being object such that
A1: x in dom F and
A2: F.x = y by FUNCT_1:def 3;
    reconsider x as Element of NATPLUS by A1;
    F.x = x(#)f by Def15;
    then y is Solution_of_Sierp168 of s by A2,Def14;
    hence thesis;
  end;
