
theorem EQCL3:
  for X be RealNormSpace,
      Y be Subset of X,
      v be object
  holds
    v in Cl(Y)
  iff
    ex seq be sequence of X st rng seq c= Y & seq is convergent & lim seq = v
  proof
    let X be RealNormSpace,
        Y be Subset of X,
        v be object;
    reconsider Z = Y as Subset of LinearTopSpaceNorm X by NORMSP_2:def 4;
    A1: Cl Z = Cl Y by EQCL1;
    hereby
      assume
      A2: v in Cl(Y); then
      A3: for G being Subset of LinearTopSpaceNorm X st G is open & v in G
      holds G meets Z by A1,PRE_TOPC:def 7;
      reconsider v0 = v as Point of X by A2;
      defpred P[Nat,Point of X] means ||.v0 -$2.|| < 1/($1 +1) & $2 in Y;
      A4: for n be Element of NAT holds ex w be Element of X st P[n, w]
      proof
        let n be Element of NAT;
        set e = 1/(n+1);
        for x be object st x in {y where y is Point of X:||.v0-y.|| < e}
        holds x in the carrier of LinearTopSpaceNorm X
        proof
          let x be object;
          assume x in {y where y is Point of X:||.v0-y.|| < e}; then
          ex y be Point of X st x = y & ||.v0 - y.|| < e; then
          x in the carrier of X;
          hence thesis by NORMSP_2:def 4;
        end; then
        reconsider G = {y where y is Point of X:||.v0-y.|| < e} as Subset of
        LinearTopSpaceNorm X by TARSKI:def 3;
        ||.v0-v0.|| < e by NORMSP_1:6; then
        v0 in G; then
        G meets Z by A3,NORMSP_2:23; then
        consider w be object such that
        A5: w in G /\ Z by XBOOLE_0:def 1;
        A6: w in G & w in Z by A5,XBOOLE_0:def 4; then
        A7: ex y be Point of X st w=y & ||.v0-y.|| < e;
        reconsider w as Point of X by A6;
        take w;
        thus thesis by A5,A7,XBOOLE_0:def 4;
      end;
      consider seq be Function of NAT, X such that
      A8: for n be Element of NAT holds P[n, seq.n] from FUNCT_2:sch 3(A4);
      take seq;
      for y be object st y in rng seq holds y in Y
      proof
        let y be object;
        assume y in rng seq; then
        ex x being object st x in NAT & seq.x = y by FUNCT_2:11;
        hence thesis by A8;
      end;
      hence rng seq c= Y;
      A10: now
        let s be Real;
        consider n being Nat such that
        A11: s" < n by SEQ_4:3;
        assume
        A12: 0 < s;
        s" + 0 < n + 1 by A11,XREAL_1:8; then
        1/(n+1) < 1/s" by A12,XREAL_1:76; then
        A13: 1/(n+1) < s by XCMPLX_1:216;
        take k = n;
        let m be Nat;
        A14: m in NAT by ORDINAL1:def 12;
        assume k <= m; then
        k+1 <= m+1 by XREAL_1:6; then
        1/(m+1) <= 1/(k+1) by XREAL_1:118; then
        1/(m+1) < s by A13,XXREAL_0:2; then
        ||.v0 -seq.m.|| < s by A8,A14,XXREAL_0:2;
        hence ||.seq.m-v0.|| <s by NORMSP_1:7;
      end;
      hence seq is convergent;
      hence lim seq = v by A10,NORMSP_1:def 7;
    end;
    given seq be sequence of X such that
    A15: rng seq c= Y and
    A16: seq is convergent and
    A17: lim seq = v;
    v in the carrier of X by A17; then
    A18: v in the carrier of (LinearTopSpaceNorm X) by NORMSP_2:def 4;
    reconsider v0 = v as Point of X by A17;
    for G being Subset of LinearTopSpaceNorm X
    st G is open & v in G holds G meets Z
    proof
      let G be Subset of LinearTopSpaceNorm X;
      assume G is open & v in G; then
      consider r be Real such that
      A20: r > 0 & {y where y is Point of X:||.v0-y.|| < r} c= G
        by NORMSP_2:22;
      consider m be Nat such that
      A21: for n be Nat st m <= n holds ||.(seq.n) - v0.|| < r
        by A16,A17,A20,NORMSP_1:def 7;
      ||.(seq.m) - v0.|| < r by A21; then
      ||.v0 - (seq.m).|| < r by NORMSP_1:7; then
      A22: seq.m in {y where y is Point of X:||.v0-y.|| < r};
      seq.m in rng seq by FUNCT_2:4,ORDINAL1:def 12;
      hence G meets Z by A15,A20,A22,XBOOLE_0:def 4;
    end;
    hence v in Cl(Y) by A1,A18,PRE_TOPC:def 7;
  end;
