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 Th7:
  seq is non-zero iff for n being Nat holds seq.n<>0.S
proof
  thus seq is non-zero implies for n being Nat holds seq.n<>0.S
   by ORDINAL1:def 12,Th6;
  assume for n being Nat holds seq.n<>0.S;
  then for x holds x in NAT implies seq.x<>0.S;
  hence thesis by Th6;
end;
