reserve i,k,l,m,n for Nat,
  x for set;
reserve R for non empty ZeroStr;
reserve p,q for AlgSequence of R;

theorem Th3:
  len p=k+1 implies p.k<>0.R
proof
  assume
A1: len p=k+1;
  then k<len p by XREAL_1:29;
  then not k is_at_least_length_of p by Def3;
  then consider i such that
A2: i>=k and
A3: p.i<>0.R;
  i<k+1 by A1,A3,Th1;
  then i<=k by NAT_1:13;
  hence thesis by A2,A3,XXREAL_0:1;
end;
