
theorem
  for X be RealUnitarySpace, f be linear-Functional of X,
      y be Point of X
  st for x be Point of X holds f.x = x .|. y holds
    f is Lipschitzian
proof
  let X be RealUnitarySpace, f be linear-Functional of X,
      y be Point of X;
  assume AS: for x be Point of X holds f.x = x .|. y;
  reconsider K=||.y.|| + 1 as Real;
A11: 0 <= ||.y.|| by BHSP_1:28;
  for x be Point of X holds |.f.x.| <= K * ||.x.||
  proof
    let x be Point of X;
B1: |.x .|. y.| <= ||.x.||*||.y.|| by BHSP_1:29;
B2: ||.y.|| < ||.y.|| + 1 by XREAL_1:145;
    0 <= ||.x.|| by BHSP_1:28; then
    ||.y.|| * ||.x.|| <= (||.y.|| + 1) * ||.x.|| by B2,XREAL_1:64; then
    |.x .|. y.| <= (||.y.|| + 1) * ||.x.|| by B1,XXREAL_0:2;
    hence thesis by AS;
  end;
  hence thesis by A11,BHSP_6:def 4;
end;
