
theorem Lm65a:
for V be RealNormSpace, x0 be Point of V st x0 <> 0.V holds
  ex G be Point of DualSp V
    st (Bound2Lipschitz(G,V)).x0 = 1 & ||.G.|| = 1/||.x0.||
proof
   let V be RealNormSpace, x0 be Point of V;
   assume AS0: x0 <> 0.V;
   set X = NLin({0.V});
   set Y = the carrier of Lin({0.V});
   for s be object holds s in Y iff s in {0.V}
   proof
    let s be object;
    hereby assume s in Y; then
     s in Lin({0.V}); then
     ex a be Real st s=a*0.V by RLVECT_4:8;
     hence s in {0.V} by TARSKI:def 1;
    end;
    assume s in {0.V}; then
    s = 1*0.V by TARSKI:def 1; then
    s in Lin{0.V} by RLVECT_4:8;
    hence s in Y;
   end; then
Y1:the carrier of X = {0.V} by TARSKI:2;
   set Z = {||.x-x0.|| where x is Point of V : x in X};
Y2:for s be object holds s in Z iff s in {||.x0.||}
   proof
    let s be object;
    hereby assume s in Z; then
     consider x be Point of V such that
Y11:  s=||.x-x0.|| & x in X;
     x = 0.V by Y1,Y11,TARSKI:def 1; then
     ||.x-x0.|| = ||.x0.|| by NORMSP_1:2;
     hence s in {||.x0.||} by TARSKI:def 1,Y11;
    end;
    assume s in {||.x0.||}; then
    s = ||.x0.|| by TARSKI:def 1; then
X1: s = ||.0.V-x0.|| by NORMSP_1:2;
    0.V in X by Y1,TARSKI:def 1;
    hence s in Z by X1;
   end; then
   reconsider Z as non empty Subset of REAL by TARSKI:2;
   reconsider d = lower_bound Z as Real;
Y3:Z = {||.x0.||} by Y2; then
X4:d = ||.x0.|| by SEQ_4:9; then
   d <> 0 by AS0,NORMSP_0:def 5; 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 X4,Th63;
   take G;
   thus thesis by X3,SEQ_4:9,Y3;
end;
