
theorem Th37:
  for K be add-associative right_zeroed right_complementable
Abelian associative well-unital distributive non empty doubleLoopStr for V,W
  be VectSp of K for v,u be Vector of V, w,t be Vector of W, f be bilinear-Form
  of V,W holds f.(v-u,w-t) = f.(v,w) - f.(v,t) -(f.(u,w) - f.(u,t))
proof
  let K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let V,W be VectSp of K;
  let v,w be Vector of V, y,z be Vector of W, f be bilinear-Form 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 Th35
    .= v1 - v3 - f.(w,y-z) by Th36
    .= v1 - v3 - (v4 - v5) by Th36;
end;
