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 Th3:
  seq is non-zero iff for n holds seq.n<>0.TOP-REAL N
proof
  thus seq is non-zero implies for n holds seq.n<>0.TOP-REAL N
    by ORDINAL1:def 12,Th2;
  assume for n holds seq.n<>0.TOP-REAL N;
  then for x holds x in NAT implies seq.x<>0.TOP-REAL N;
  hence thesis by Th2;
end;
