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;
reserve p for Element of QC-WFF(A);

theorem Th16:
  p is universal implies len @the_scope_of p < len @p
proof
  assume
A1: p is universal;
  then consider x being bound_QC-variable of A, q being Element of QC-WFF(A)
       such that
A2: p = All(x, q);
  len @q + len <*x*> = len (<*x*>^@q) by FINSEQ_1:22;
  then
A3: len @q <= len (<*x*>^@q) by NAT_1:11;
  p = <*[3, 0]*>^(<*x*>^@q) by A2,FINSEQ_1:32;
  then len @p = len <*[3, 0]*> + len (<*x*>^@q) by FINSEQ_1:22
    .= len (<*x*>^@q) + 1 by FINSEQ_1:40;
  then len @q < len @p by A3,NAT_1:13;
  hence thesis by A1,A2,Def28;
end;
