
theorem Th16:
  for V being RealLinearSpace, M being non empty Affine Subset of
V, v being VECTOR of V st v in M holds ex N being non empty Affine Subset of V
  st N = M - {v} & M is_parallel_to N & N is Subspace-like
proof
  let V be RealLinearSpace;
  let M be non empty Affine Subset of V;
  let v be VECTOR of V;
  {v} is non empty Affine by RUSUB_4:23;
  then reconsider N = M - {v} as non empty Affine Subset of V by Th4,Th8;
  assume v in M;
  then
A1: 0.V in N by Th15;
  take N;
  M is_parallel_to N
  proof
    take v;
    thus thesis by Th9;
  end;
  hence thesis by A1,RUSUB_4:26;
end;
