
theorem Th33:
  for K be add-associative right_zeroed right_complementable
  associative left_unital distributive non empty doubleLoopStr for V,W be
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital non empty
ModuleStr over K for f be homogeneousSAF Form of V,W, w be Vector of W holds f
  .(0.V,w) = 0.K
proof
  let F be add-associative right_zeroed right_complementable associative
  left_unital distributive non empty doubleLoopStr;
  let V,W be add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
  non empty ModuleStr over F;
  let f be homogeneousSAF Form of V,W, v be Vector of W;
  thus f.(0.V,v) = f.((0.F)*(0.V),v) by VECTSP10:1
    .= 0.F *f.(0.V,v) by Th31
    .= 0.F;
end;
