
theorem HTh37:
  for V, W being Z_Module, v, u being Vector of V, w, t being Vector of W,
  f being bilinear-FrForm of V,W holds
  f.(v-u,w-t) = f.(v,w) - f.(v,t) -(f.(u,w) - f.(u,t))
  proof
    let V, W be Z_Module;
    let v, w be Vector of V, y, z be Vector of W, f be bilinear-FrForm of V,W;
    set v1 = f.(v,y), v3 = f.(v,z), v4 = f.(w,y), v5 = f.(w,z);
    thus f.(v-w,y-z) = f.(v,y-z) - f.(w,y-z) by HTh35
    .= v1 - v3 - f.(w,y-z) by HTh36
    .= v1 - v3 - (v4 - v5) by HTh36;
  end;
