
theorem RLSUB134:
  for V be RealNormSpace,
      V1 be SubRealNormSpace of V,
      S be Subset of V
  st S = the carrier of V1
  holds S is linearly-closed
  proof
    let V be RealNormSpace,
        V1 be SubRealNormSpace of V,
        S be Subset of V;
    assume
    A1: S = the carrier of V1;
    A2: for v,u be VECTOR of V st v in S & u in S holds v+u in S
    proof
      let v,u be VECTOR of V such that
      A3: v in S and
      A4: u in S;
      reconsider v1 = v, u1 = u as Point of V1 by A1,A3,A4;
      v+u = v1+u1 by SUBTH0;
      hence thesis by A1;
    end;
    for r be Real for v be VECTOR of V st v in S holds r*v in S
    proof
      let r be Real;
      let v be VECTOR of V such that
      A5: v in S;
      reconsider v1 = v as Point of V1 by A1,A5;
      r*v = r*v1 by SUBTH0;
      hence thesis by A1;
    end;
    hence thesis by A2;
  end;
