
theorem VS10Th28:
  for V being non trivial free Z_Module,
  f being non constant 0-preserving FrFunctional of V
  ex v being Vector of V st v <> 0.V & f.v <> 0.F_Real
  proof
    let V be non trivial free Z_Module,
    f be non constant 0-preserving FrFunctional of V;
    A1: f.(0.V) = 0.F_Real by HDef9;
    assume
    A2: for v being Vector of V st v <> 0.V holds f.v = 0.F_Real;
    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 by A2,A1
      .= f.w by A2,A1
      .= f.y;
    end;
    hence contradiction by FUNCT_1:def 10;
  end;
