
theorem
  for V, W being Z_Module, v, u being Vector of V, w, t being Vector of W,
  a, b being Element of INT.Ring, f being bilinear-FrForm of V,W
  holds f.(v-a*u,w-b*t) = f.(v,w) - b*f.(v,t) - (a*f.(u,w) - a*(b*f.(u,t)))
  proof
    let V, W be Z_Module, v, w be Vector of V, y, z be Vector of W,
    a, b be Element of INT.Ring, 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-a*w,y-b*z) = v1 -f.(v,b*z) - (f.(a*w,y) -f.(a*w,b*z)) by HTh37
    .= v1 -b*v3 - (f.(a*w,y) -f.(a*w,b*z)) by HTh32
    .= v1 - b*v3 - (a*v4 - f.(a*w,b*z)) by HTh31
    .= v1 - b*v3 - (a*v4 - a*f.(w,b*z)) by HTh31
    .= v1 - b*v3 - (a*v4 - a*(b*v5)) by HTh32;
  end;
