reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;

theorem
  A is boundary iff for F being Subset of X holds F is closed implies
  Int F = Int(F \/ A)
proof
  thus A is boundary implies for F being Subset of X holds F is closed implies
  Int F = Int(F \/ A)
  proof
    assume
A1: Int A = {};
    let F be Subset of X;
    assume F is closed;
    then Int(F \/ A) = Int(F \/ Int A) by Th6;
    hence thesis by A1;
  end;
  assume
  for F being Subset of X holds F is closed implies Int F = Int(F \/ A );
  then Int {}X = Int({}X \/ A);
  hence thesis;
end;
