reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve a for Element of A;

theorem Th63:
  card bound_QC-variables(Al) = card QC-symbols(Al) &
  not bound_QC-variables(Al) is finite
proof
  NAT c= QC-symbols(Al) by QC_LANG1:3;
  then
A1: not QC-symbols(Al) is finite;
  bound_QC-variables(Al) = [: {4}, QC-symbols(Al) :] by QC_LANG1:def 4;
  then card bound_QC-variables(Al) = card [:QC-symbols(Al),{4}:] by CARD_2:4;
  hence thesis by A1,CARD_4:19;
end;
