
theorem
  for V being RealLinearSpace, M be non empty Affine Subset of V holds
  ex L being non empty Affine Subset of V st L = M - M & L is Subspace-like & M
  is_parallel_to L
proof
  let V be RealLinearSpace;
  let M be non empty Affine Subset of V;
  reconsider L = M - M as non empty Affine Subset of V by Th4,Th8;
  consider v being object such that
A1: v in M by XBOOLE_0:def 1;
  reconsider v as VECTOR of V by A1;
  take L;
A2: 0.V in L by Th12;
  {v} is non empty Affine by RUSUB_4:23;
  then reconsider N = M - {v} as non empty Affine Subset of V by Th4,Th8;
A3: M is_parallel_to N
  proof
    take v;
    thus thesis by Th9;
  end;
  N = union {M - {u} where u is VECTOR of V : u in M} by A1,Th19
    .= L by Th18;
  hence thesis by A3,A2,RUSUB_4:26;
end;
