
theorem Lm89A:
  for X be RealUnitarySpace, f be linear-Functional of X
    st f is Lipschitzian
  holds f"{0} is closed
proof
  let X be RealUnitarySpace, f be linear-Functional of X;
  assume AS: f is Lipschitzian;
  set Y=f"{0};
  for s be sequence of X st rng s c= Y & s is convergent
    holds lim s in Y
  proof
    let s be sequence of X;
    assume B0: rng s c= Y & s is convergent;
    reconsider x0=lim s as Point of X;
B1: dom f = the carrier of X by FUNCT_2:def 1;
    consider K be Real such that
B3:   0 < K &
       for x be Point of X
         holds |. f.x .| <= K * ||. x .|| by AS,BHSP_6:def 4;
    for x1, x2 be Point of X
      st x1 in the carrier of X & x2 in the carrier of X
     holds |.f/.x1 - f/.x2.| <= K * ||.x1 - x2.||
    proof
      let x1,x2 be Point of X;
      assume x1 in the carrier of X & x2 in the carrier of X;
C1:   |.f/.x1 - f/.x2.| = |.f.(x1 - x2).| by HAHNBAN:19;
      thus thesis by C1,B3;
    end; then
    f is_Lipschitzian_on (the carrier of X) by FUNCT_2:def 1,B3; then
B41: f is_continuous_on (the carrier of X) by LM5;
B5: rng s c= the carrier of X;
B71: f is_continuous_in x0 by B41;
B91: f/*s = f*s by B1,B5,FUNCT_2:def 11;
    ex k be Nat st
      for n be Nat st k <= n holds (f*s).n = (seq_const 0).n
    proof
      set k = the Nat;
C0:   for n be Nat st k <= n holds (f*s).n = (seq_const 0).n
      proof
        let n be Nat;
        assume k <= n;
        s.n in rng s by FUNCT_2:4,ORDINAL1:def 12; then
D2:     s.n in X & f.(s.n) in {0} by FUNCT_2:38,B0;
        dom s = NAT by FUNCT_2:def 1; then
        (f*s).n in {0} by ORDINAL1:def 12,D2,FUNCT_1:13; then
        (f*s).n = 0 by TARSKI:def 1;
        hence (f*s).n = (seq_const 0).n by SEQ_1:57;
      end;
      take k;
      thus thesis by C0;
    end; then
    lim (f*s) = lim (seq_const 0) by SEQ_4:19
             .= (seq_const 0).0 by SEQ_4:26
             .= 0; then
    f.x0 = 0 by B71,B0,B1,B91; then
    x0 in X & f.x0 in {0} by TARSKI:def 1;
    hence lim s in Y by FUNCT_2:38;
  end;
  hence f"{0} is closed by LM1;
end;
