reserve A for QC-alphabet;
reserve a,b,b1,b2,c,d for object,
  i,j,k,n for Nat,
  x,y,x1,x2 for bound_QC-variable of A,
  P for QC-pred_symbol of k,A,
  ll for CQC-variable_list of k,A,
  l1 ,l2 for FinSequence of QC-variables(A),
  p for QC-formula of A,
  s,t for QC-symbol of A;
reserve Sub for CQC_Substitution of A;
reserve finSub for finite CQC_Substitution of A;
reserve e for Element of vSUB(A);
reserve S,S9,S1,S2,S19,S29,T1,T2 for Element of QC-Sub-WFF(A);
reserve B for Element of [:QC-Sub-WFF(A),bound_QC-variables(A):];
reserve SQ for second_Q_comp of B;

theorem Th24:
  S is Sub_universal implies len@((Sub_the_scope_of(S))`1) < len @ (S`1)
proof
  assume S is Sub_universal;
  then consider B,SQ such that
A1: S = Sub_All(B,SQ) & B is quantifiable;
  S = [All(B`2,(B`1)`1),SQ] by A1,Def24;
  then
A2: S`1 = All(B`2,(B`1)`1);
  All(B`2,(B`1)`1) is universal;
  then
A3: len @the_scope_of All(B`2,(B`1)`1) < len @(S`1) by A2,QC_LANG1:16;
  (Sub_the_scope_of(S))`1 = (B`1)`1 by A1,Th21;
  hence thesis by A3,QC_LANG2:7;
end;
