
theorem Lm65:
for V be RealNormSpace, x0 be Point of V st x0 <> 0.V holds
  ex F be Point of DualSp V
     st ||.F.|| = 1 & (Bound2Lipschitz(F,V)).x0 =||.x0.||
proof
   let V be RealNormSpace, x0 be Point of V;
   assume AS: x0 <> 0.V; then
   consider G be Point of DualSp V such that
A2: (Bound2Lipschitz(G,V)).x0 = 1 & ||.G.|| = 1/||.x0.|| by Lm65a;
   reconsider d=||.x0.|| as Real;
   reconsider F=d*G as Point of DualSp V;
   take F;
A4: ||.F.|| = |.d.|*||.G.|| by NORMSP_1:def 1
           .= d*(1/d) by A2,ABSVALUE:def 1
           .= 1 by AS,NORMSP_0:def 5,XCMPLX_1:106;
   (Bound2Lipschitz(F,V)).x0
        = d * G.x0 by DUALSP01:30,SUBSET_1:def 8
       .= d *(Bound2Lipschitz(G,V)).x0 by SUBSET_1:def 8;
   hence thesis by A2,A4;
end;
