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 Th4:
  len p = len q & (for k st k < len p holds p.k = q.k) implies p=q
proof
  assume that
A1: len p = len q and
A2: for k st k < len p holds p.k = q.k;
A3: for x being object st x in NAT holds p.x=q.x
  proof
    let x be object;
    assume x in NAT;
    then reconsider k=x as Element of NAT;
    k >= len p implies p.k = q.k
    proof
      assume
A4:   k >= len p;
      then p.k = 0.R by Th1;
      hence thesis by A1,A4,Th1;
    end;
    hence thesis by A2;
  end;
  dom p = NAT & dom q = NAT by FUNCT_2:def 1;
  hence thesis by A3,FUNCT_1:2;
end;
