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

theorem Th24:
  for X being set, T being 1-sorted, F being ManySortedSet of X
  holds F is net_set of X,T iff for i being set st i in X holds F.i is net of T
proof
  let X be set, T be 1-sorted, F be ManySortedSet of X;
  hereby
    assume
A1: F is net_set of X,T;
    let i be set;
    assume i in X;
    then i in dom F by PARTFUN1:def 2;
    then F.i in rng F by FUNCT_1:def 3;
    hence F.i is net of T by A1,Def12;
  end;
  assume
A2: for i being set st i in X holds F.i is net of T;
  let x be set;
  assume x in rng F;
  then ex i being object st i in dom F & x = F.i by FUNCT_1:def 3;
  hence thesis by A2;
end;
