
theorem
  for V,W being add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
  non empty ModuleStr over INT.Ring, 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 add-associative right_zeroed right_complementable
    vector-distributive scalar-distributive scalar-associative scalar-unital
    non empty ModuleStr over INT.Ring,
    v, w be Vector of V, y, z be Vector of W, a, b be Element of INT.Ring;
    let 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 HTh28
    .= 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;
