theorem Th9:
  (for m holds p.m = q.m) implies p = q
proof
  assume
A1: for m holds p.m = q.m;
  for j be Nat st j in Seg(n+1) holds p.j = q.j
  proof
    let j be Nat;
    assume j in Seg(n+1);
    then reconsider j as Nat of n by Th7;
    p.j = q.j by A1;
    hence thesis;
  end;
  hence thesis by FINSEQ_2:119;
end;
