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 Th30:
  S is Sub_conjunctive implies CQC_Sub(S) = (CQC_Sub(
  Sub_the_left_argument_of S)) '&' (CQC_Sub(Sub_the_right_argument_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 F2 being Function of QC-Sub-WFF(A),QC-WFF(A) such that
A3: CQC_Sub(Sub_the_right_argument_of S) = F2.(Sub_the_right_argument_of S) and
A4: for S9 being Element of QC-Sub-WFF(A) holds (S9 is A-Sub_VERUM implies F2
.S9 = VERUM(A)) & ( S9 is Sub_atomic implies F2.S9 =
(the_pred_symbol_of ((S9)`1))
! CQC_Subst(Sub_the_arguments_of S9,(S9)`2)) & (S9 is Sub_negative implies F2.
S9 = 'not' (F2.(Sub_the_argument_of S9))) & (S9 is Sub_conjunctive implies F2.
S9 = (F2.Sub_the_left_argument_of S9) '&' (F2.Sub_the_right_argument_of S9)) &
(S9 is Sub_universal implies F2.S9 = Quant(S9,F2.Sub_the_scope_of S9)) by Def38
  ;
A5: F2 = F by A2,A4,Lm6;
  consider F1 being Function of QC-Sub-WFF(A),QC-WFF(A) such that
A6: CQC_Sub(Sub_the_left_argument_of S) = F1.(Sub_the_left_argument_of S ) and
A7: for S9 being Element of QC-Sub-WFF(A) holds (S9 is A-Sub_VERUM implies F1
.S9 = VERUM(A)) & ( S9 is Sub_atomic implies F1.S9 =
(the_pred_symbol_of ((S9)`1))
! CQC_Subst(Sub_the_arguments_of S9,(S9)`2)) & (S9 is Sub_negative implies F1.
S9 = 'not' (F1.(Sub_the_argument_of S9))) & (S9 is Sub_conjunctive implies F1.
S9 = (F1.Sub_the_left_argument_of S9) '&' (F1.Sub_the_right_argument_of S9)) &
(S9 is Sub_universal implies F1.S9 = Quant(S9,F1.Sub_the_scope_of S9)) by Def38
  ;
  F1 = F by A2,A7,Lm6;
  hence thesis by A1,A2,A6,A3,A5;
end;
