reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;

theorem Th6:
  seq is non-zero iff for x st x in NAT holds seq.x<>0.S
proof
  thus seq is non-zero implies for x st x in NAT holds seq.x<>0.S
  proof
    assume seq is non-zero;
    then
A1: rng seq c= NonZero S;
    let x;
    assume x in NAT;
    then x in dom seq by FUNCT_2:def 1;
    then seq.x in rng seq by FUNCT_1:def 3;
    then not seq.x in {0.S} by A1,XBOOLE_0:def 5;
    hence thesis by TARSKI:def 1;
  end;
  assume
A2: for x st x in NAT holds seq.x<>0.S;
  now
    let y be object;
    assume
A3: y in rng seq;
    then ex x being object st x in dom seq & seq.x=y by FUNCT_1:def 3;
    then y<>0.S by A2;
    then not y in {0.S} by TARSKI:def 1;
    hence y in NonZero S by A3,XBOOLE_0:def 5;
  end;
  then rng seq c= NonZero S;
  hence thesis;
end;
