
theorem Cl01:
  for X be RealNormSpace,
      Y be Subspace of X,
      CY be Subset of X
  st CY = the carrier of Y
  holds Cl(CY) is linearly-closed
  proof
    let X be RealNormSpace,
        Y be Subspace of X,
        CY be Subset of X;
    assume
    A1: CY = the carrier of Y;
    A2: for v,u be Point of X st v in Cl(CY) & u in Cl(CY) holds v+u in Cl(CY)
    proof
      let v,u be Point of X;
      assume
      A3: v in Cl(CY) & u in Cl(CY);
      thus v+u in Cl(CY)
      proof
        consider seqv be sequence of X such that
        A4: rng seqv c= CY & seqv is convergent & lim seqv = v by A3,EQCL3;
        consider sequ be sequence of X such that
        A5: rng sequ c= CY & sequ is convergent & lim sequ = u by A3,EQCL3;
        now
          let y be object;
          assume y in rng (seqv + sequ); then
          consider x being object such that
          A6: x in NAT & (seqv + sequ).x = y by FUNCT_2:11;
          reconsider x as Element of NAT by A6;
          seqv.x in rng seqv & sequ.x in rng sequ by FUNCT_2:4; then
          seqv.x in Y & sequ.x in Y by A1,A4,A5; then
          seqv.x + sequ.x in Y by RLSUB_1:20;
          hence y in CY by A1,A6,NORMSP_1:def 2;
        end; then
        A7: rng(seqv + sequ) c= CY;
        lim(seqv + sequ) = v+u by A4,A5,NORMSP_1:25;
        hence v+u in Cl(CY) by A4,A5,A7,EQCL3,NORMSP_1:19;
      end;
    end;
    for r be Real, v be Point of X st v in Cl(CY) holds r*v in Cl(CY)
    proof
      let r be Real, v be Point of X;
      assume
      A8: v in Cl(CY);
      thus r*v in Cl(CY)
      proof
        consider seqv be sequence of X such that
        A9: rng seqv c= CY & seqv is convergent & lim seqv = v by A8,EQCL3;
        A10: r*seqv is convergent & lim(r*seqv) = r*lim(seqv)
        by A9,NORMSP_1:22,NORMSP_1:28;
        now
          let y be object;
          assume y in rng (r*seqv); then
          consider x being object such that
          A11: x in NAT & (r*seqv).x = y by FUNCT_2:11;
          reconsider x as Element of NAT by A11;
          seqv.x in rng seqv by FUNCT_2:4; then
          seqv.x in Y by A1,A9; then
          r*seqv.x in Y by RLSUB_1:21;
          hence y in CY by A1,A11,NORMSP_1:def 5;
        end; then
        rng(r*seqv) c= CY;
        hence r*v in Cl(CY) by A9,A10,EQCL3;
      end;
    end;
    hence thesis by A2;
  end;
