reserve a,b,c for set;

theorem Th26:
  for X,x0 being set st x0 in X for A being Subset of
  DiscrWithInfin(X,x0) st A is infinite holds Cl A = A \/ {x0}
proof
  let X,x0 be set such that
A1: x0 in X;
  set T = DiscrWithInfin(X,x0);
  reconsider T as non empty TopSpace by A1;
  reconsider x = x0 as Point of T by A1,Def5;
  let A be Subset of DiscrWithInfin(X,x0);
  reconsider B = {x} as Subset of T;
  reconsider A9 = A as Subset of T;
  x0 in {x0} by TARSKI:def 1;
  then x0 in A9 \/ B by XBOOLE_0:def 3;
  then A9\/B is closed by Th20;
  then
A2: Cl(A9\/B) = A9\/B by PRE_TOPC:22;
  assume A is infinite;
  then A9 is not closed or x0 in A by A1,Th20;
  hence thesis by A2,Th25,ZFMISC_1:40;
end;
