reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem Th5:
  for Y be non empty set for f being sequence of Y holds rng f
  = {f.k : 0 <= k}
proof
  let Y be non empty set;
  let f be sequence of Y;
  set Z = {f.k : 0 <= k};
A1: dom f = NAT by FUNCT_2:def 1;
A2: rng f c= Z
  proof
    let y be object;
    assume y in rng f;
    then consider n be object such that
A3: n in NAT and
A4: y = f.n by A1,FUNCT_1:def 3;
    reconsider n as Element of NAT by A3;
    0 <= n by NAT_1:2;
    hence thesis by A4;
  end;
  Z c= rng f
  proof
    let x be object;
    assume x in Z;
    then consider n1 being Nat such that
A5:   x=f.n1 & 0 <= n1;
     n1 in NAT by ORDINAL1:def 12;
    hence thesis by FUNCT_2:4,A5;
  end;
  hence thesis by A2,XBOOLE_0:def 10;
end;
