
theorem Th34:
  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 homogeneousFAF Form of V,W, v be Vector of V holds f
  .(v,0.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 homogeneousFAF Form of V,W, v be Vector of V;
  thus f.(v,0.W) = f.(v,(0.F)*(0.W)) by VECTSP10:1
    .= 0.F * f.(v,0.W) by Th32
    .= 0.F;
end;
