
theorem Th20:
  for K be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr for V,W be non empty ModuleStr
  over K for f be Functional of V,v be Vector of V, w be Vector of W holds
  FormFunctional(f,0Functional(W)).(v,w) = 0.K
proof
  let K be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr;
  let V,W be non empty ModuleStr over K;
  let f be Functional of V, v be Vector of V, y be Vector of W;
  set 0F = 0Functional(W), F = FormFunctional(f,0F);
  thus F.(v,y) = f.v * 0F.y by Def10
    .= f.v * 0.K by FUNCOP_1:7
    .= 0.K;
end;
