
theorem Th14:
  for V being RealLinearSpace, M being non empty Affine Subset of
  V, N1,N2 being non empty Affine Subset of V st N1 is Subspace-like & N2 is
  Subspace-like & M is_parallel_to N1 & M is_parallel_to N2 holds N1 = N2
proof
  let V be RealLinearSpace;
  let M,N1,N2 be non empty Affine Subset of V;
  assume that
A1: N1 is Subspace-like and
A2: N2 is Subspace-like and
A3: M is_parallel_to N1 and
A4: M is_parallel_to N2;
  N2 is_parallel_to M by A4,Th2;
  then N2 is_parallel_to N1 by A3,Th3;
  then consider v2 being VECTOR of V such that
A5: N2 = N1 + {v2};
  N1 is_parallel_to M by A3,Th2;
  then N1 is_parallel_to N2 by A4,Th3;
  then consider v1 being VECTOR of V such that
A6: N1 = N2 + {v1};
  0.V in N2 by A2,RUSUB_4:def 7;
  then 0.V in {p + q where p,q is Element of V : p in N1 & q in {v2}} by A5,
RUSUB_4:def 9;
  then consider p2,q2 being Element of V such that
A7: 0.V = p2 + q2 and
A8: p2 in N1 and
A9: q2 in {v2};
  0.V = p2 + v2 by A7,A9,TARSKI:def 1;
  then
A10: -v2 in N1 by A8,RLVECT_1:6;
  v2 = -(-v2)
    .= (-1) * (-v2) by RLVECT_1:16;
  then v2 in N1 by A1,A10,RUSUB_4:def 7;
  then
A11: N2 c= N1 by A1,A5,Th13;
  0.V in N1 by A1,RUSUB_4:def 7;
  then 0.V in {p + q where p,q is Element of V : p in N2 & q in {v1}} by A6,
RUSUB_4:def 9;
  then consider p1,q1 being Element of V such that
A12: 0.V = p1 + q1 and
A13: p1 in N2 and
A14: q1 in {v1};
  0.V = p1 + v1 by A12,A14,TARSKI:def 1;
  then
A15: -v1 in N2 by A13,RLVECT_1:6;
  v1 = -(-v1)
    .= (-1) * (-v1) by RLVECT_1:16;
  then v1 in N2 by A2,A15,RUSUB_4:def 7;
  then N1 c= N2 by A2,A6,Th13;
  hence thesis by A11;
end;
