reserve f for Function;
reserve n,k,n1 for Nat;
reserve r,p for Real;
reserve x,y,z for object;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Real_Sequence;

theorem Th4:
  seq is non-zero iff for x st x in NAT holds seq.x<>0
proof
  thus seq is non-zero implies for x st x in NAT holds seq.x<>0
  proof
    assume
A1: seq is non-zero;
    let x;
    assume x in NAT;
    then x in dom seq by Th2;
    then seq.x in rng seq by FUNCT_1:def 3;
    hence thesis by A1;
  end;
  assume
A2: for x st x in NAT holds seq.x<>0;
  assume 0 in rng seq;
  then ex x being object st x in dom seq & seq.x=0 by FUNCT_1:def 3;
  hence contradiction by A2;
end;
