reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;

theorem Th12:
  A is open iff Int A = A
  proof
    hereby
      assume
A1:   A is open;
A2:   Int A = union {O where O is open Subset of NT: O c= A} by Lm5;
      now
        let o be object;
        assume
A3:     o in A;
        A in {O where O is open Subset of NT: O c= A} by A1;
        hence o in Int A by A2,A3,TARSKI:def 4;
      end;
      then A c= Int A;
      hence Int A = A by Lm15;
    end;
    assume Int A = A;
    hence A is open;
  end;
