reserve Al for QC-alphabet;
reserve i,j,n,k,l for Nat;
reserve a for set;
reserve T,S,X,Y for Subset of CQC-WFF(Al);
reserve p,q,r,t,F,H,G for Element of CQC-WFF(Al);
reserve s for QC-formula of Al;
reserve x,y for bound_QC-variable of Al;
reserve f,g for FinSequence of [:CQC-WFF(Al),Proof_Step_Kinds:];

theorem
  p in Cn(X) implies ex Y st Y c= X & Y is finite & p in Cn(Y)
proof
  assume p in Cn(X);
  then consider f such that
A1: f is_a_proof_wrt X and
A2: Effect(f) = p by Th32;
A3: f <> {} by A1;
 consider A being set such that
A4: A is finite and
A5: A c= CQC-WFF(Al) and
A6: rng f c= [:A,Proof_Step_Kinds:] by FINSEQ_1:def 4,FINSET_1:14;
  reconsider Z=A as Subset of CQC-WFF(Al) by A5;
  take Y = Z /\ X;
  thus Y c= X by XBOOLE_1:17;
  thus Y is finite by A4;
 1 <= n & n <= len f implies f,n is_a_correct_step_wrt Y
  proof
    assume
A7: 1 <= n & n <= len f;
then A8: f,n is_a_correct_step_wrt X by A1;
    (f.n)`2 = 0 or ... or  (f.n)`2 = 9 by A7,Th19;
    then per cases;
    case
  (f.n)`2 = 0;
then A9:  (f.n)`1 in X by A8,Def4;
  n in Seg(len f) by A7,FINSEQ_1:1;
then   n in dom f by FINSEQ_1:def 3;
then   f.n in rng f by FUNCT_1:def 3;
then   (f.n)`1 in A by A6,MCART_1:10;
      hence thesis by A9,XBOOLE_0:def 4;
    end;
    case
  (f.n)`2 = 1;
      hence thesis by A8,Def4;
    end;
    case
  (f.n)`2 = 2;
      hence thesis by A8,Def4;
    end;
    case
  (f.n)`2 = 3;
      hence thesis by A8,Def4;
    end;
    case
  (f.n)`2 = 4;
      hence thesis by A8,Def4;
    end;
    case
  (f.n)`2 = 5;
      hence thesis by A8,Def4;
    end;
    case
  (f.n)`2 = 6;
      hence thesis by A8,Def4;
    end;
    case
  (f.n)`2 = 7;
      hence thesis by A8,Def4;
    end;
    case
  (f.n)`2 = 8;
      hence thesis by A8,Def4;
    end;
    case
  (f.n)`2 = 9;
      hence thesis by A8,Def4;
    end;
  end;
then  f is_a_proof_wrt Y by A3;
then  p in {q: ex f st f is_a_proof_wrt Y & Effect(f) = q} by A2;
  hence thesis by Th31;
end;
