
theorem Th13:
  for V being RealLinearSpace, M being non empty Affine Subset of
  V, v being VECTOR of V st M is Subspace-like & v in M holds M + {v} c= M
proof
  let V be RealLinearSpace;
  let M be non empty Affine Subset of V;
  let v be VECTOR of V;
  assume
A1: M is Subspace-like & v in M;
    let x be object;
    assume
A2: x in M + {v};
    then reconsider x as Element of V;
    x in {u1 + v1 where u1,v1 is Element of V : u1 in M & v1 in {v}} by A2,
RUSUB_4:def 9;
    then consider u1,v1 being Element of V such that
A3: x = u1 + v1 & u1 in M and
A4: v1 in {v};
    v1 = v by A4,TARSKI:def 1;
    hence thesis by A1,A3,RUSUB_4:def 7;
end;
