reserve x for Real;
reserve p,q for Point of TOP-REAL 2;

theorem Th20:
  for D being non empty Subset of TOP-REAL 2 st D=NonZero TOP-REAL
  2 holds D`= {0.TOP-REAL 2}
proof
  let D be non empty Subset of TOP-REAL 2;
  assume
A1: D=NonZero TOP-REAL 2;
A2: D` c= {0.TOP-REAL 2}
  proof
    let x be object;
    assume
A3: x in D`;
    then x in (the carrier of TOP-REAL 2)\D by SUBSET_1:def 4;
    then not x in D by XBOOLE_0:def 5;
    hence thesis by A1,A3,XBOOLE_0:def 5;
  end;
  {0.TOP-REAL 2} c= D`
  proof
    let x be object;
    assume
A4: x in {0.TOP-REAL 2};
    then not x in D by A1,XBOOLE_0:def 5;
    then x in (the carrier of TOP-REAL 2)\D by A4,XBOOLE_0:def 5;
    hence thesis by SUBSET_1:def 4;
  end;
  hence thesis by A2;
end;
