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 Th10:
  for F being Element of QC-WFF(A) holds 1 <= len @F
proof
  let F be Element of QC-WFF(A);
  now
    per cases by Th9;
    suppose
      F = VERUM(A);
      hence thesis by FINSEQ_1:39;
    end;
    suppose
      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
A1:   F = p!ll;
      @F = <*p*>^ll by A1,Th8;
      then len @F = len <*p*> + len ll by FINSEQ_1:22
        .= 1 + len ll by FINSEQ_1:39;
      hence thesis by NAT_1:11;
    end;
    suppose
      F is negative;
      then consider p being Element of QC-WFF(A) such that
A2:   F = 'not' p;
      len @F = len <*[1, 0]*> + len @p by A2,FINSEQ_1:22
        .= 1 + len @p by FINSEQ_1:39;
      hence thesis by NAT_1:11;
    end;
    suppose
      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 len @F = len <*[2, 0]*> + len (@p^@q) by FINSEQ_1:22
        .= 1 + len (@p^@q) by FINSEQ_1:39;
      hence thesis by NAT_1:11;
    end;
    suppose
      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 len @F = len <*[3, 0]*> + len (<*x*>^@p) by FINSEQ_1:22
        .= 1 + len (<*x*>^@p) by FINSEQ_1:39;
      hence thesis by NAT_1:11;
    end;
  end;
  hence thesis;
end;
