
theorem Th3:
  for V being Abelian add-associative right_zeroed
  right_complementable non empty RLSStruct, M,L,N be Affine Subset of V st M
  is_parallel_to L & L is_parallel_to N holds M is_parallel_to N
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty RLSStruct;
  let M,L,N be Affine Subset of V;
  assume that
A1: M is_parallel_to L and
A2: L is_parallel_to N;
  consider v1 being Element of V such that
A3: M = L + {v1} by A1;
  consider u1 being Element of V such that
A4: L = N + {u1} by A2;
  set w = u1 + v1;
  for x being object st x in N + {w} holds x in M
  proof
    let x be object;
A5: u1 in {u1} by TARSKI:def 1;
    assume
A6: x in N + {w};
    then reconsider x as Element of V;
    x in {u + v where u,v is Element of V: u in N & v in {w}} by A6,
RUSUB_4:def 9;
    then consider u2,v2 being Element of V such that
A7: x = u2 + v2 and
A8: u2 in N and
A9: v2 in {w};
    x = u2 + (u1 + v1) by A7,A9,TARSKI:def 1
      .= u2 + u1 + v1 by RLVECT_1:def 3;
    then x - v1 = u2 + u1 + (v1 - v1) by RLVECT_1:def 3
      .= u2 + u1 + 0.V by RLVECT_1:15
      .= u2 + u1;
    then x - v1 in {u + v where u,v is Element of V : u in N & v in {u1}} by A8
,A5;
    then
A10: x - v1 in L by A4,RUSUB_4:def 9;
    set y = x - v1;
A11: v1 in {v1} by TARSKI:def 1;
    y + v1 = x - (v1 - v1) by RLVECT_1:29
      .= x - 0.V by RLVECT_1:15
      .= x;
    then
    x in {u + v where u,v is Element of V: u in L & v in {v1}} by A10,A11;
    hence thesis by A3,RUSUB_4:def 9;
  end;
  then
A12: N + {w} c= M;
  for x being object st x in M holds x in N + {w}
  proof
    let x be object;
A13: w in {w} by TARSKI:def 1;
    assume
A14: x in M;
    then reconsider x as Element of V;
    x in {u + v where u,v is Element of V: u in L & v in {v1} } by A3,A14,
RUSUB_4:def 9;
    then consider u2,v2 being Element of V such that
A15: x = u2 + v2 and
A16: u2 in L and
A17: v2 in {v1};
A18: v2 = v1 by A17,TARSKI:def 1;
    u2 in {u + v where u,v is Element of V: u in N & v in {u1 } } by A4,A16,
RUSUB_4:def 9;
    then consider u3,v3 being Element of V such that
A19: u2 = u3 + v3 and
A20: u3 in N and
A21: v3 in {u1};
    v3 = u1 by A21,TARSKI:def 1;
    then x = u3 + w by A15,A19,A18,RLVECT_1:def 3;
    then
    x in {u + v where u,v is Element of V: u in N & v in {w}} by A20,A13;
    hence thesis by RUSUB_4:def 9;
  end;
  then M c= N + {w};
  then M = N + {w} by A12;
  hence thesis;
end;
