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);

theorem Th8:
  ex p,e st S = [p,e]
proof
  [:QC-WFF(A),vSUB(A):] is Subset of [:[:NAT, QC-symbols(A):]*,vSUB(A):] &
(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*>^ll,e]
in [:QC-WFF(A),vSUB(A):]) & (for e
  being Element of vSUB(A) holds [<*[0, 0]*>,e] in
[:QC-WFF(A),vSUB(A):]) & (for p being
FinSequence of [:NAT,QC-symbols(A):], e being Element of vSUB(A)
st [p,e] in [:QC-WFF(A),vSUB(A):]
holds [<*[1, 0]*>^p,e] in [:QC-WFF(A),vSUB(A):]) &
(for p, q being FinSequence of [:
  NAT, QC-symbols(A):], e being Element of vSUB(A) st
[p,e] in [:QC-WFF(A),vSUB(A):] & [q,e] in [:
  QC-WFF(A),vSUB(A):] holds [<*[2, 0]*>^p^q,e]
in [:QC-WFF(A),vSUB(A):]) & (for x being
bound_QC-variable of A, p being FinSequence of [:NAT, QC-symbols(A):],
e being Element of vSUB(A)
st [p,(QSub(A)).[<*[3, 0]*>^<*x*>^p,e]] in [:QC-WFF(A),vSUB(A):]
holds [<*[3, 0]*>^<*x*>^p,e] in [:QC-WFF(A),vSUB(A):]) by Th7;
  then [:QC-WFF(A),vSUB(A):] is A-Sub-closed;
  then QC-Sub-WFF(A) c= [:QC-WFF(A),vSUB(A):] by Def17;
  then S in [:QC-WFF(A),vSUB(A):];
  then consider a,b being object such that
A1: a in QC-WFF(A) and
A2: b in vSUB(A) and
A3: S = [a,b] by ZFMISC_1:def 2;
  reconsider e = b as Element of vSUB(A) by A2;
  reconsider p = a as Element of QC-WFF(A) by A1;
  take p,e;
  thus thesis by A3;
end;
