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);
reserve Q for QC-pred_symbol of A;
reserve F, G for (Element of QC-WFF(A)), s for FinSequence;

theorem Th12:
  ((@F.1)`1 = 0 implies F = VERUM(A)) & ((@F.1)`1 = 1 implies F is
  negative) & ((@F.1)`1 = 2 implies F is conjunctive) & ((@F.1)`1 = 3 implies F
  is universal) & ((ex k being Nat
                     st @F.1 is QC-pred_symbol of k, A)
  implies F is atomic)
proof
A1: now
    per cases by Th9;
    case
      F is atomic;
      then consider
      k being Nat, P being (QC-pred_symbol of k, A), ll being
      QC-variable_list of k, A such that
A2:   F = P!ll;
      @F = <*P*>^ll by A2,Th8;
      then @F.1 = P by FINSEQ_1:41;
      hence ex k being Nat st @F.1 is QC-pred_symbol of k, A;
    end;
    case
      F = VERUM(A);
      hence (@F.1)`1 = [0,0]`1
        .= 0;
    end;
    case
      F is negative;
      then ex p being Element of QC-WFF(A) st F = 'not' p;
      then @F.1 = [1, 0] by FINSEQ_1:41;
      hence (@F.1)`1 = 1;
    end;
    case
      F is conjunctive;
      then consider p, q being Element of QC-WFF(A) such that
A3:   F = p '&' q;
      @F = <*[2, 0]*>^(@p^@q) by A3,FINSEQ_1:32;
      then @F.1 = [2, 0] by FINSEQ_1:41;
      hence (@F.1)`1 = 2;
    end;
    case
      F is universal;
      then consider
      x being bound_QC-variable of A, p being Element of QC-WFF(A) such that
A4:   F = All(x, p);
      @F = <*[3, 0]*>^(<*x*>^@p) by A4,FINSEQ_1:32;
      then @F.1 = [3, 0] by FINSEQ_1:41;
      hence (@F.1)`1 = 3;
    end;
  end;
  now
    let k be Nat, P be QC-pred_symbol of k, A;
    reconsider P9 = P as QC-pred_symbol of A;
    P9`1 = 7+the_arity_of P9 by Def8;
    hence P`1 <> 0 & P`1 <> 1 & P`1 <> 2 & P`1 <> 3 by NAT_1:11;
  end;
  hence thesis by A1;
end;
