
theorem
  for X be RealHilbertSpace, f be Point of DualSp X,
      g be Lipschitzian linear-Functional of X
  st g=f holds
   ex y be Point of X st
     (for x be Point of X holds g.x = x .|. y)
   & ||.f.|| = ||.y.||
proof
  let X be RealHilbertSpace, f be Point of DualSp X,
      g be Lipschitzian linear-Functional of X;
  assume AS: g=f;
  consider y be Point of X such that
A1: for x be Point of X holds g.x = x .|. y by Th89A;
  now let s be Real;
    assume s in PreNorms g; then
    consider t be VECTOR of X such that
B1:   s = |.g.t.| & ||.t.|| <= 1;
B3: |.t .|. y.| <= ||.t.|| * ||.y.|| by BHSP_1:29;
    0 <= ||.y.|| by BHSP_1:28; then
    ||.t.|| * ||.y.|| <= 1 * ||.y.|| by B1,XREAL_1:64; then
    |.t .|. y.| <= ||.y.|| by B3,XXREAL_0:2;
    hence s <= ||.y.|| by B1,A1;
  end; then
  upper_bound PreNorms g <= ||.y.|| by SEQ_4:45; then
A2: ||.f.|| <= ||.y.|| by AS,Th24;
A31: ||.y.|| <= ||.f.||
  proof
    per cases;
    suppose ||.y.|| = 0;
      hence ||.y.|| <= ||.f.|| by Th27;
    end;
    suppose AS2: ||.y.|| <> 0;
B1:   0 <= y .|. y by BHSP_1:def 2;
B2:   g.y = y .|. y by A1
         .= ||.y.||^2 by B1,SQUARE_1:def 2
         .= ||.y.|| * ||.y.|| by SQUARE_1:def 1;
B3:   g.y <= |.g.y.| by ABSVALUE:4;
      |.g.y.| <= ||.f.|| * ||.y.|| by AS,Th26; then
B4:   ||.y.|| * ||.y.|| <= ||.f.|| * ||.y.|| by B2,B3,XXREAL_0:2;
B51:  0 <= ||.y.|| by BHSP_1:28;
      ||.y.|| * ||.y.||/||.y.|| = ||.y.|| &
      ||.f.|| * ||.y.||/||.y.|| = ||.f.|| by AS2,XCMPLX_1:89;
      hence ||.y.|| <= ||.f.|| by B51,B4,XREAL_1:72;
    end;
  end;
  take y;
  thus thesis by A1,A2,XXREAL_0:1,A31;
end;
