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;

theorem Th20:
  Al is countable implies
  ExCl(Al) is non empty & ExCl(Al) is countable
proof
  assume A1: Al is countable;
  set x =the  bound_QC-variable of Al,p =the  Element of CQC-WFF(Al);
  set a = Ex(x,p);
  a in ExCl(Al) by Def3;
  hence ExCl(Al) is non empty;
  CQC-WFF(Al) is countable by Th19,A1;
  hence thesis;
end;
