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 Th17:
  seq is non-zero implies seq ^\k is non-zero
proof
  assume
A1: seq is non-zero;
  now
    let n be Nat;
    (seq ^\k).n=seq.(n+k) by NAT_1:def 3;
    hence (seq ^\k).n<>0.S by A1,Th7;
  end;
  hence thesis by Th7;
end;
