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 Th32:
  S is Sub_universal implies CQC_Sub(S) = Quant(S,CQC_Sub( Sub_the_scope_of S))
proof
  consider F being Function of QC-Sub-WFF(A),QC-WFF(A) such that
A1: CQC_Sub(S) = F.S and
A2: 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;
  consider G being Function of QC-Sub-WFF(A),QC-WFF(A) such that
A3: CQC_Sub(Sub_the_scope_of S) = G.(Sub_the_scope_of S) and
A4: for S9 being Element of QC-Sub-WFF(A) holds (S9 is A-Sub_VERUM implies G.
S9 = VERUM(A)) & ( S9 is Sub_atomic implies G.S9 =
(the_pred_symbol_of ((S9)`1))!
CQC_Subst(Sub_the_arguments_of S9,(S9)`2)) & (S9 is Sub_negative implies G.S9 =
'not' (G.(Sub_the_argument_of S9))) & (S9 is Sub_conjunctive implies G.S9 = (G.
  Sub_the_left_argument_of S9) '&' (G.Sub_the_right_argument_of S9)) & (S9 is
  Sub_universal implies G.S9 = Quant(S9,G.Sub_the_scope_of S9)) by Def38;
  F = G by A2,A4,Lm6;
  hence thesis by A1,A2,A3;
end;
