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
  f is_a_proof_wrt X implies (f.1)`2 = 0 or ... or (f.1)`2 = 6
proof
  assume
A1: f is_a_proof_wrt X;
then A2: 1 <= len f by Th21;
then A3: f,1 is_a_correct_step_wrt X by A1;
  assume
  A4: (f.1)`2 <> 0 & ... & (f.1)`2 <> 6;
  (f.1)`2 = 0 or ... or (f.1)`2 = 9 by A2,Th19;
  then per cases by A4;
  suppose
 (f.1)`2 = 7;
then  ex i,j,p,q st 1 <= i & i < 1 & 1 <= j & j < i &
    p = (f.j)`1 & q = (f.1)`1 & (f.i)`1 = p => q by A3,Def4;
    hence contradiction;
  end;
  suppose
 (f.1)`2 = 8;
then  ex i,p,q,x st 1 <= i & i < 1 & (f.i)`1 = p => q &
    not x in still_not-bound_in p & (f.1)`1 = p => All(x,q) by A3,Def4;
    hence contradiction;
  end;
  suppose
 (f.1)`2 = 9;
then  ex i,x,y,s st 1 <= i & i < 1 &
    s.x in CQC-WFF(Al) & s.y in CQC-WFF(Al) & not x in still_not-bound_in s &
    s.x = (f.i)`1 & (f.1)`1 = s.y by A3,Def4;
    hence contradiction;
  end;
end;
