
theorem Lm73:
  for X be strict RealNormSpace, A be non empty Subset of X holds
    (for f be Point of DualSp X st
     (for x be Point of X st x in A holds (Bound2Lipschitz(f,X)).x = 0)
     holds Bound2Lipschitz(f,X) = 0.(DualSp X))
   implies ClNLin(A) = X
proof
  let X be strict RealNormSpace, A be non empty Subset of X;
  assume
A0: for f be Point of DualSp X st
      (for x be Point of X st x in A holds (Bound2Lipschitz(f,X)).x = 0)
    holds Bound2Lipschitz(f,X) = 0.(DualSp X);
    set M = ClNLin(A);
    consider Z be Subset of X such that
Q0:   Z = the carrier of Lin(A)
    & M = NORMSTR(# Cl(Z), Zero_(Cl(Z),X), Add_(Cl(Z),X),
                    Mult_(Cl(Z),X), Norm_(Cl(Z),X) #) by NORMSP_3:def 20;
    reconsider Y = the carrier of M as non empty Subset of X
      by DUALSP01:def 16;
Q23: Y is linearly-closed by NORMSP_3:29;
    Y = the carrier of X
    proof
      assume Y <> the carrier of X; then
      not the carrier of X c= Y; then
      consider x0 be object such that
Q25:    x0 in the carrier of X & not x0 in Y;
      reconsider x0 as Point of X by Q25;
      consider G be Point of DualSp X such that
Q26:    (for x be Point of X
           st x in Y holds (Bound2Lipschitz(G,X)).x = 0) and
Q27:    (Bound2Lipschitz(G,X)).x0 = 1 by Q0,Q23,Q25,DUALSP02:2;
      for x be Point of X
        st x in A holds (Bound2Lipschitz(G,X)).x = 0
      proof
        let x be Point of X;
        assume x in A; then
        x in Lin(A) by RLVECT_3:15; then
        x in Y by Q0,NORMSP_3:4,TARSKI:def 3;
        hence (Bound2Lipschitz(G,X)).x = 0 by Q26;
      end; then
      (Bound2Lipschitz(G,X)).x0 = (0.(DualSp X)).x0 by A0
                               .= ((the carrier of X) --> 0).x0 by DUALSP01:25
                               .= 0;
      hence contradiction by Q27;
    end;
    hence M = X by Q0,NORMSP_3:26;
end;
