
theorem
  for V being right_zeroed non empty RLSStruct, M be Affine Subset of
  V holds M is_parallel_to M
proof
  let V be right_zeroed non empty RLSStruct;
  let M be Affine Subset of V;
  take 0.V;
  for x being object st x in M + {0.V} holds x in M
  proof
    let x be object;
    assume x in M + {0.V};
    then x in {u + v where u,v is Element of V: u in M & v in {0.V}} by
RUSUB_4:def 9;
    then consider u,v being Element of V such that
A1: x = u + v & u in M and
A2: v in {0.V};
    v = 0.V by A2,TARSKI:def 1;
    hence thesis by A1,RLVECT_1:def 4;
  end;
  then
A3: M + {0.V} c= M;
  for x being object st x in M holds x in M + {0.V}
  proof
    let x be object;
    assume
A4: x in M;
    then reconsider x as Element of V;
    x = x + 0.V & 0.V in {0.V} by RLVECT_1:def 4,TARSKI:def 1;
    then x in {u + v where u,v is Element of V: u in M & v in {0.V}} by A4;
    hence thesis by RUSUB_4:def 9;
  end;
  then M c= M + {0.V};
  hence thesis by A3;
end;
