reserve A for QC-alphabet;
reserve k,n,m for Nat;
reserve P for QC-pred_symbol of A;
reserve F for Element of QC-WFF(A);

theorem Th9:
  for F being Element of QC-WFF(A) holds F = VERUM(A) or F is atomic or
  F is negative or F is conjunctive or F is universal
proof
  defpred P[Element of QC-WFF(A)] means $1 = VERUM(A) or $1 is atomic or $1 is
  negative or $1 is conjunctive or $1 is universal;
A1: P[VERUM(A)];
A2: for p being Element of QC-WFF(A) st P[p] holds P['not' p] by Def19;
A3: for x being bound_QC-variable of A,
        p being Element of QC-WFF(A) st P[p] holds P[All(x, p)] by Def21;
A4: for p, q being Element of QC-WFF(A)
     st P[p] & P[q] holds P[p '&' q] by Def20;
A5: for k being Nat, p being (QC-pred_symbol of k, A), ll being
  QC-variable_list of k,A holds P[p!ll] by Def18;
  thus for F being Element of QC-WFF(A)
             holds P[F] from QCInd (A5, A1, A2, A4, A3);
end;
