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;
reserve Z for Element of [:QC-WFF(A),vSUB(A):];

theorem Th35:
  CQC_Sub(Sub_P(P,ll,e)) is Element of CQC-WFF(A)
proof
  set l = Sub_the_arguments_of Sub_P(P,ll,e);
A1: l is CQC-variable_list of k,A by Def29;
  then reconsider l as FinSequence of bound_QC-variables(A) by Th34;
  reconsider s = CQC_Subst(l,Sub_P(P,ll,e)`2) as FinSequence of
  bound_QC-variables(A);
  len l = k by A1,CARD_1:def 7;
  then
A2: len s = k by Def3;
  Sub_P(P,ll,e) = [P!ll,e] by Th9;
  then Sub_P(P,ll,e)`1 = P!ll;
  then reconsider
  P9 = the_pred_symbol_of Sub_P(P,ll,e)`1 as QC-pred_symbol of k,A by Lm7;
  reconsider s as CQC-variable_list of k,A by A2,Th34;
  ex F being Function of QC-Sub-WFF(A),QC-WFF(A)
  st CQC_Sub(Sub_P (P,ll,e)) = F.Sub_P(P,ll,e) &
  for S9 being Element of QC-Sub-WFF(A)
  holds (S9 is A-Sub_VERUM implies F. S9 = VERUM(A)) &
  ( S9 is Sub_atomic implies F.S9 = ( the_pred_symbol_of
  ((S9)`1))! CQC_Subst(Sub_the_arguments_of S9,(S9)`2)) & (S9 is Sub_negative
  implies F.S9 = 'not' (F.(Sub_the_argument_of S9))) & (S9 is Sub_conjunctive
  implies F.S9 = (F. Sub_the_left_argument_of S9) '&' (F.
Sub_the_right_argument_of S9)) & (S9 is Sub_universal implies F.S9 = Quant(S9,F
  .Sub_the_scope_of S9)) by Def38;
  then CQC_Sub(Sub_P(P,ll,e)) = P9!s;
  hence thesis;
end;
