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;
