
theorem Th39:
  for V be add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over F_Complex for f be cmplxhomogeneousFAF
  Form of V,V for v be Vector of V holds f.(v,0.V) = 0.F_Complex
proof
  let V be add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over F_Complex;
  let f be cmplxhomogeneousFAF Form of V,V, v be Vector of V;
  set F=FunctionalFAF(f,v);
  thus f.(v,0.V) = f.(v,0.F_Complex *0.V) by VECTSP10:1
    .= F.(0.F_Complex *0.V) by BILINEAR:8
    .= (0.F_Complex)*' * F.(0.V) by Def1
    .= 0.F_Complex by COMPLFLD:47;
end;
