
theorem
  for X be RealUnitarySpace, M be Subspace of X,
      N be non empty Subset of X
   st N = the carrier of (Ort_Comp M) holds
   N is linearly-closed & N is closed
proof
  let X be RealUnitarySpace, M be Subspace of X,
      N be non empty Subset of X;
  assume AS1: N = the carrier of (Ort_Comp M);
  hence N is linearly-closed by RUSUB_1:28;
  for s be sequence of X st rng s c= N & s is convergent
      holds lim s in N
  proof
    let s be sequence of X;
    assume AS2: rng s c= N & s is convergent;
A1: now let i be Nat;
      s.i in rng s by FUNCT_2:4,ORDINAL1:def 12; then
      s.i in N by AS2; then
      s.i in {v where v is VECTOR of X : for w being VECTOR of X st
                w in M holds w,v are_orthogonal} by AS1,RUSUB_5:def 3; then
      consider v be VECTOR of X such that
B1:     v=s.i & for w being VECTOR of X st w in M holds w,v are_orthogonal;
      thus for w being VECTOR of X st w in M holds w .|. (s.i) = 0
        by B1,BHSP_1:def 3;
    end;
    for w being VECTOR of X st w in M holds w .|. (lim s) = 0
    proof
      let w be VECTOR of X;
      assume AS3: w in M;
      reconsider g=w .|. (lim s) as Real;
      for p be Real st 0 < p ex m be Nat st
        for n be Nat st m <= n holds |. (seq_const 0).n - w .|. (lim s) .| < p
      proof
        let p be Real;
        assume B0: 0 < p;
B1:     0 <= ||.w.|| by BHSP_1:28;
        reconsider r=p/(||.w.|| + 1) as Real;
B41:    ||.w.|| + 0 < ||.w.|| + 1 by XREAL_1:8;
        r*(||.w.|| + 1) = p by B1,XCMPLX_1:87; then
B5:     0 < r & r*||.w.|| < p by B0,B1,B41,XREAL_1:68;
        consider m be Nat such that
B6:       for n be Nat st m <= n holds ||.s.n - lim s.|| < r
            by B1,B0,AS2,BHSP_2:19;
B7:     for n be Nat st m <= n holds |. (seq_const 0).n - w .|. (lim s) .| < p
        proof
          let n be Nat;
          assume m <= n; then
C1:       ||.s.n - lim s.|| < r by B6;
C2:       |. w .|. (s.n) - w .|. (lim s) .| = |. w .|. (s.n - lim s) .|
            by BHSP_1:12;
C3:       |. w .|. (s.n - lim s) .| <= ||.w.|| * ||.s.n - lim s.||
            by BHSP_1:29;
          ||.w.|| * ||.s.n - lim s.|| <= ||.w.||*r by B1,C1,XREAL_1:64; then
C41:      |. w .|. (s.n) - w .|. (lim s) .| <= ||.w.||*r
            by C2,C3,XXREAL_0:2;
          w .|. (s.n) = 0 by A1,AS3
                     .= (seq_const 0).n by SEQ_1:57;
          hence thesis by C41,B5,XXREAL_0:2;
        end;
        take m;
        thus thesis by B7;
      end; then
      lim (seq_const 0) = w .|. (lim s) by SEQ_2:def 7; then
      (seq_const 0).0 = w .|. (lim s) by SEQ_4:26;
      hence w .|. (lim s) = 0;
    end; then
A3: for w being VECTOR of X st w in M holds w,lim s are_orthogonal;
    reconsider v=lim s as VECTOR of X;
    lim s in {v where v is VECTOR of X : for w being VECTOR of X st
                w in M holds w,v are_orthogonal} by A3;
    hence lim s in N by AS1,RUSUB_5:def 3;
  end;
  hence N is closed by LM1;
end;
