
theorem Lm64:
for V be RealNormSpace, Y be non empty Subset of V,
    x0 be Point of V
  st Y is linearly-closed & Y is closed & not x0 in Y holds
   ex G be Point of DualSp V st
    (for x be Point of V st x in Y holds (Bound2Lipschitz(G,V)).x = 0 )
  & (Bound2Lipschitz(G,V)).x0 = 1
proof
   let V be RealNormSpace, Y be non empty Subset of V,
       x0 be Point of V;
   assume AS: Y is linearly-closed & Y is closed & not x0 in Y;
   set X = NLin(Y);
X1:the carrier of X = Y by NORMSP_3:31,AS;
   set Z = {||.x-x0.|| where x is Point of V : x in X};
   X is Subspace of V by NORMSP_3:27; then
   0.V in X by RLSUB_1:17; then
X2: ||.0.V-x0.|| in Z;
   now let z be object;
    assume z in Z; then
    ex x be Point of V st z = ||.x-x0.|| & x in X;
    hence z in REAL;
   end; then
   Z c= REAL; then
   reconsider Z as non empty Subset of REAL by X2;
   reconsider r0 = 0 as Real;
   for r be ExtReal st r in Z holds r0 <= r
   proof
    let r be ExtReal;
    assume r in Z; then
    ex x be Point of V st r = ||.x-x0.|| & x in X;
    hence r0 <= r;
   end; then
U1:r0 is LowerBound of Z by XXREAL_2:def 2; then
   Z is bounded_below; then
   reconsider Z as non empty bounded_below real-membered
     Subset of REAL;
   reconsider d = lower_bound Z as Real;
X3:r0 <= inf Z by U1,XXREAL_2:def 4;
   d > 0
   proof
    assume not d > 0; then
X22:d = 0 by X3;
    reconsider Yt = Y` as Subset of TopSpaceNorm V;
Z24:Yt is open by AS,NORMSP_2:16;
    x0 in (the carrier of V) \ Y by AS,XBOOLE_0:def 5; then
    x0 in Yt by SUBSET_1:def 4; then
    consider s be Real such that
X23:  0 < s
    & {y where y is Point of V : ||.x0-y.|| < s} c= Yt by Z24,NORMSP_2:7;
    consider r be Real such that
X24:  r in Z & r < 0 + s by X22,X23,SEQ_4:def 2;
    consider x be Point of V such that
X25:  r = ||.x-x0.|| & x in X by X24;
    ||.x0-x.||< s by X24,X25,NORMSP_1:7; then
    x in {x where x is Point of V : ||.x0-x.|| < s}; then
    x in Yt by X23; then
    x in (the carrier of V) \ Y by SUBSET_1:def 4;
    hence contradiction by X1,X25,XBOOLE_0:def 5;
   end; then
   consider G be Point of DualSp V such that
X3: ( for x be Point of V st x in X holds (Bound2Lipschitz(G,V)).x = 0 )
  & (Bound2Lipschitz(G,V)).x0 = 1 & ||.G.|| = 1/d by Th63;
   take G;
   now let x be Point of V;
    assume x in Y; then
    x in X by NORMSP_3:31,AS;
    hence (Bound2Lipschitz(G,V)).x = 0 by X3;
   end;
   hence thesis by X3;
end;
