
theorem Th2:
  for V being add-associative right_zeroed right_complementablenon
  empty RLSStruct, M,N be Affine Subset of V st M is_parallel_to N holds N
  is_parallel_to M
proof
  let V be add-associative right_zeroed right_complementable non empty
  RLSStruct;
  let M,N be Affine Subset of V;
  assume M is_parallel_to N;
  then consider w1 being VECTOR of V such that
A1: M = N + {w1};
  set w2 = - w1;
  for x being object st x in N holds x in M + {w2}
  proof
    let x be object;
    assume
A2: x in N;
    then reconsider x as Element of V;
    set y = x + w1;
    w1 in {w1} by TARSKI:def 1;
    then y in {u + v where u,v is Element of V: u in N & v in {w1} } by A2;
    then
A3: y in M by A1,RUSUB_4:def 9;
    x + (w1 + w2) = y + w2 by RLVECT_1:def 3;
    then x + 0.V = y + w2 by RLVECT_1:5;
    then
A4: x = y + w2;
    w2 in {w2} by TARSKI:def 1;
    then x in {u + v where u,v is Element of V: u in M & v in {w2}} by A3,A4;
    hence thesis by RUSUB_4:def 9;
  end;
  then
A5: N c= M + {w2};
  take w2;
  for x being object st x in M + {w2} holds x in N
  proof
    let x be object;
    assume
A6: x in M + {w2};
    then x in {u + v where u,v is Element of V: u in M & v in {w2}} by
RUSUB_4:def 9;
    then consider u9,v9 being Element of V such that
A7: x = u9 + v9 and
A8: u9 in M and
A9: v9 in {w2};
    reconsider x as Element of V by A6;
    x = u9 + w2 by A7,A9,TARSKI:def 1;
    then x + w1 = u9 + (w2 + w1) by RLVECT_1:def 3;
    then
A10: x + w1 = u9 + 0.V by RLVECT_1:5;
    u9 in {u + v where u,v is Element of V: u in N & v in {w1 } } by A1,A8,
RUSUB_4:def 9;
    then consider u1,v1 being Element of V such that
A11: u9 = u1 + v1 & u1 in N and
A12: v1 in {w1};
    w1 =v1 by A12,TARSKI:def 1;
    hence thesis by A10,A11,RLVECT_1:8;
  end;
  then M + {w2} c= N;
  hence thesis by A5;
end;
