
theorem Th1:
  for T being T_1 non empty TopSpace, A being finite Subset of T
  holds A is closed
proof
  let T be T_1 non empty TopSpace, A be finite Subset of T;
  defpred P[set] means ex S being Subset of T st $1 = S & S is closed;
A1: P[{}]
  proof
    take {}T;
    thus {}T = {};
    thus thesis;
  end;
A2: for x, C being set st x in A & C c= A & P[C] holds P[C \/ {x}]
  proof
    let x, C be set;
    assume that
A3: x in A and
    C c= A and
A4: P[C];
    reconsider y = x as Element of T by A3;
    consider S being Subset of T such that
A5: C = S and
A6: S is closed by A4;
    {y} is closed by URYSOHN1:19;
    then S \/ {y} is closed by A6;
    hence thesis by A5;
  end;
A7: A is finite;
  P[A] from FINSET_1:sch 2(A7,A1,A2);
  hence thesis;
end;
