theorem Th31:
  for n being Nat holds 1 <= n & n <= len PR implies
    (PR.n)`2 = 0 or ... or (PR.n)`2 = 9
proof
  let n be Nat;
  assume 1 <= n & n <= len PR;
  then n in dom PR by FINSEQ_3:25;
  then PR.n in rng PR by FUNCT_1:def 3;
  then (PR.n)`2 in {k where k is Nat: k <= 9} by CQC_THE1:def 3,MCART_1:10;
  then ex k being Nat st k = (PR.n)`2 & k <= 9;
  hence thesis;
end;
