reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;

theorem
  for A being Subset of X holds A is anti-discrete & A is not open
  implies A is boundary
proof
  let A be Subset of X;
A1: Int A c= A by TOPS_1:16;
  assume A is anti-discrete;
  then A misses Int A or A c= Int A;
  then
A2: A /\ Int A = {} or A c= Int A;
  assume
A3: A is not open;
  assume A is not boundary;
  then Int A <> {} by TOPS_1:48;
  hence contradiction by A3,A2,A1,XBOOLE_0:def 10,XBOOLE_1:28;
end;
