reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;
reserve Nseq for increasing sequence of NAT;

theorem
  seq is sequence of NAT iff for n holds seq.n is Element of NAT
proof
  thus seq is sequence of NAT implies for n holds seq.n is Element of NAT
  by ORDINAL1:def 12;
  assume
A1: for n holds seq.n is Element of NAT;
  rng seq c= NAT
  proof
    let x be object;
    assume x in rng seq;
    then consider y being object such that
A2: y in dom seq and
A3: x = seq.y by FUNCT_1:def 3;
    reconsider y as Element of NAT by A2,FUNCT_2:def 1;
    x = seq.y by A3;
    then x is Element of NAT by A1;
    hence thesis;
  end;
  hence thesis by FUNCT_2:6;
end;
