theorem
  f is sequence of N iff dom f = NAT & for n holds f.n is Element of N
proof
  thus f is sequence of N implies
    dom f = NAT & for n holds f.n is Element of N
  by Lm1,ORDINAL1:def 12;
  assume that
A1: dom f = NAT and
A2: for n holds f.n is Element of N;
  for x holds x in NAT implies f.x is Element of N by A2;
  hence thesis by A1,Lm1;
end;
