reserve N for Nat;
reserve n,m,n1,n2 for Nat;
reserve q,r,r1,r2 for Real;
reserve x,y for set;
reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N;
reserve seq,seq1,seq2,seq3,seq9 for Real_Sequence of N;

theorem Th1:
  for f being Function holds f is Real_Sequence of N iff (dom f=NAT
  & for n holds f.n is Point of TOP-REAL N)
proof
  let f be Function;
  thus f is Real_Sequence of N implies
  (dom f=NAT & for n being Nat holds f.n is Point
  of TOP-REAL N)
  by NORMSP_1:12,ORDINAL1:def 12;
  assume that
A1: dom f=NAT and
A2: for n holds f.n is Point of TOP-REAL N;
  for x holds x in NAT implies f.x is Point of TOP-REAL N by A2;
  hence thesis by A1,NORMSP_1:12;
end;
