reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve a for Element of A;

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;
