reserve Al for QC-alphabet;
reserve b,c,d for set,
  X,Y for Subset of CQC-WFF(Al),
  i,j,k,m,n for Nat,
  p,p1,q,r,s,s1 for Element of CQC-WFF(Al),
  x,x1,x2,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al, A,
  v for Element of Valuations_in(Al,A),
  f1,f2 for FinSequence of CQC-WFF(Al),
  CX,CY,CZ for Consistent Subset of CQC-WFF(Al),
  JH for Henkin_interpretation of CX,
  a for Element of A,
  t,u for QC-symbol of Al;
reserve L for PATH of q,p,
  F1,F3 for QC-formula of Al,
  a for set;
reserve C,D for Element of [:CQC-WFF(Al),bool bound_QC-variables(Al):];
reserve K,L for Subset of bound_QC-variables(Al);

theorem Th28:
  for A being Subset of bound_QC-variables(Al) st A is finite holds
  ex x st not x in A
proof
  let A be Subset of bound_QC-variables(Al);
  A1: not bound_QC-variables(Al) is finite by CALCUL_1:64;
  assume A is finite;
  then not (for b being object holds b in A iff b in bound_QC-variables(Al))
  by A1,TARSKI:2;
  then consider b such that
A2: not b in A and
A3: b in bound_QC-variables(Al);
  reconsider x = b as bound_QC-variable of Al by A3;
  take x;
  thus thesis by A2;
end;
