reserve i,k for Nat;
reserve A for QC-alphabet;
reserve x for bound_QC-variable of A;
reserve a for free_QC-variable of A;
reserve p,q for Element of QC-WFF(A);
reserve l for FinSequence of QC-variables(A);
reserve P,Q for QC-pred_symbol of A;
reserve V for non empty Subset of QC-variables(A);
reserve s,t for QC-symbol of A;

theorem Th9:
  for p being QC-formula of A st p is conjunctive holds
  still_not-bound_in p = (still_not-bound_in the_left_argument_of p) \/ (
  still_not-bound_in the_right_argument_of p)
proof
  deffunc F1(Element of QC-WFF(A)) = still_not-bound_in $1;
  deffunc A1(Element of QC-WFF(A)) =
           still_not-bound_in(the_arguments_of $1);
  deffunc Q1(Element of QC-WFF(A),
             Subset of bound_QC-variables(A)) = $2 \ {bound_in $1};
  deffunc N1(Subset of bound_QC-variables(A)) = $1;
  deffunc C1(Subset of bound_QC-variables(A),
             Subset of bound_QC-variables(A)) = $1 \/ $2;
  A1: for p being QC-formula of A, d being Subset of bound_QC-variables(A)
     holds d = F1(p) iff ex F being Function of QC-WFF(A),
     (bool bound_QC-variables(A)) st d = F.p &
for p being Element of QC-WFF(A)
for d1,d2 being Subset of bound_QC-variables(A)
holds (p = VERUM(A) implies F.p = ({} bound_QC-variables(A))) &
(p is atomic implies F.p = A1(p)) & (p is negative &
d1 = F.the_argument_of p implies F.p = N1(d1))
& (p is conjunctive & d1 = F.the_left_argument_of p & d2 = F.
the_right_argument_of p implies F.p = C1(d1,d2)) &
(p is universal & d1 = F.the_scope_of p implies F.p = Q1(p,d1))
by Lm1;
  let p be QC-formula of A such that
A2: p is conjunctive;
  for d1,d2 being Subset of bound_QC-variables(A) st d1 = F1(
the_left_argument_of p) & d2 = F1(the_right_argument_of p) holds F1(p) = C1(d1,
  d2) from QCDResult9conjunctive(A1,A2);
  hence thesis;
end;
