reserve a,b,c for set;

theorem Th25:
  for T being non empty TopSpace for A being Subset of T st A is
  not closed for a being Point of T st A \/ {a} is closed holds Cl A = A \/ {a}
proof
  let T be non empty TopSpace;
  let A be Subset of T such that
A1: A is not closed;
A2: A c= Cl A by PRE_TOPC:18;
  let a be Point of T;
  assume A \/ {a} is closed;
  then
A3: Cl (A\/{a}) = A\/{a} by PRE_TOPC:22;
  Cl A c= Cl (A\/{a}) by PRE_TOPC:19,XBOOLE_1:7;
  hence thesis by A1,A2,A3,ZFMISC_1:138;
end;
