
theorem
  for X be RealUnitarySpace, f be linear-Functional of X holds
    for y1,y2 be Point of X st
      for x be Point of X holds f.x = x .|. y1 & f.x = x .|. y2
     holds y1 = y2
proof
  let X be RealUnitarySpace, f be linear-Functional of X;
  let y1,y2 be Point of X;
  assume AS: for x be Point of X holds f.x = x .|. y1 & f.x = x .|. y2;
    now let x be Point of X;
      f.x = x .|. y1 & f.x = x .|. y2 by AS; then
      x .|. y1 - x .|. y2 = 0;
      hence x .|. (y1 - y2) = 0 by BHSP_1:12;
    end; then
    (y1 - y2) .|. (y1 - y2) = 0; then
    y1 - y2 = 0.X by BHSP_1:def 2;
    hence y1 = y2 by RLVECT_1:21;
end;
