
theorem Th28:
  for K be Field, V be non trivial VectSp of K for f be non
constant 0-preserving Functional of V ex v be Vector of V st v <> 0.V & f.v <>
  0.K
proof
  let K be Field, V be non trivial VectSp of K, f be non constant 0-preserving
  Functional of V;
A1: f.(0.V) =0.K by HAHNBAN1:def 9;
  assume
A2: for v be Vector of V st v <> 0.V holds f.v = 0.K;
  now
    let x,y be object;
    assume x in dom f & y in dom f;
    then reconsider v=x,w=y as Vector of V;
    thus f.x = f.v .= 0.K by A2,A1
      .= f.w by A2,A1
      .= f.y;
  end;
  hence contradiction by FUNCT_1:def 10;
end;
