reserve a,b,c for set;

theorem Th20:
  for X,x0 being set, A being Subset of DiscrWithInfin(X,x0) holds
  A is closed iff (x0 in X implies x0 in A) or A is finite
proof
  let X,x0 be set;
  set T = DiscrWithInfin(X,x0);
  let A be Subset of DiscrWithInfin(X,x0);
A1: A is closed iff not x0 in A` or A`` is finite by Th19;
  the carrier of T = X by Def5;
  hence thesis by A1,XBOOLE_0:def 5;
end;
