 reserve x for Real,
    p,k,l,m,n,s,h,i,j,k1,t,t1 for Nat,
    X for Subset of REAL;
reserve x for object, X,Y,Z for set;
 reserve M,N for Cardinal;
reserve X for non empty set,
  s for sequence of X;

theorem
  (for n holds s.n in Y) implies rng s c= Y
proof
  assume
A1: for n holds s.n in Y;
  let y be object;
  assume y in rng s;
  then consider x being object such that
A2: x in dom s and
A3: y = s.x by FUNCT_1:def 3;
  x in NAT by A2,FUNCT_2:def 1;
  hence thesis by A1,A3;
end;
