reserve A for QC-alphabet;
reserve a,b,b1,b2,c,d for object,
  i,j,k,n for Nat,
  x,y,x1,x2 for bound_QC-variable of A,
  P for QC-pred_symbol of k,A,
  ll for CQC-variable_list of k,A,
  l1 ,l2 for FinSequence of QC-variables(A),
  p for QC-formula of A,
  s,t for QC-symbol of A;
reserve Sub for CQC_Substitution of A;
reserve finSub for finite CQC_Substitution of A;
reserve e for Element of vSUB(A);
reserve S,S9,S1,S2,S19,S29,T1,T2 for Element of QC-Sub-WFF(A);
reserve B for Element of [:QC-Sub-WFF(A),bound_QC-variables(A):];
reserve SQ for second_Q_comp of B;

theorem Th12:
  for S holds S is A-Sub_VERUM or S is Sub_atomic or S is
  Sub_negative or S is Sub_conjunctive or S is Sub_universal
proof
  defpred P[Element of QC-Sub-WFF(A)] means
         $1 is A-Sub_VERUM or $1 is Sub_atomic
  or $1 is Sub_negative or $1 is Sub_conjunctive or $1 is Sub_universal;
A1: for k being Nat, p being (QC-pred_symbol of k,A), ll being
  QC-variable_list of k,A, e being Element of vSUB(A) holds P[Sub_P(p,ll,e)];
A2: for S being Element of QC-Sub-WFF(A) st S is A-Sub_VERUM holds P[S];
A3: for x being bound_QC-variable of A, S being Element of QC-Sub-WFF(A),
   SQ being
  second_Q_comp of [S,x] st [S,x] is quantifiable & P[S] holds P[Sub_All([S,x],
  SQ)] by Def28;
A4: for S1, S2 being Element of QC-Sub-WFF(A) st S1`2 = S2`2 & P[S1] & P[S2]
  holds P[Sub_&(S1,S2)] by Def27;
A5: for S being Element of QC-Sub-WFF(A) st P[S] holds P[Sub_not(S)] by Def26;
  thus for S being Element of QC-Sub-WFF(A) holds
   P[S] from SubQCInd (A1, A2, A5,
  A4, A3);
end;
