Can someone please help me with this proof I dont know how t

Can someone please help me with this proof? I dont know how to go about this one. Thanks! :)

Let v+W be a subspace of V

So let w be in W

Hence,v+w is in v+W

v+W is subspace so

-v-w is in v+W

Hence, for some x in W

v+x=-v-w

Adding v to both sides

2v+x=-w

2v=-x-w

HEnce, v is in W

Now let v in W

So for any x,y in v+W

x+y is in W since v+W is in W

So, x+y=v+(-v+x+y) which is in v+W

Hence, v+W is a subspace

v1+W=v2+W means for any

v1+w in v1+W there is some v2+v in v2+W so that

v1+w=v2+v

Hence,v1-v2=v-w

But v-w is in W

Hence, v1-v2 is in W

Let, v1-v2 be in W

Hence

v1-v2=w for some w in W

So for any v1+x in v1+W for x in W

v1+x=v1-v2+v2+x=v2+(w+x) which is v2+W

Now let ,v2+y be in v2+W for y in W

v2+y=v2-v1+v1+y=v1+(-w+y) which is in v1+W

Hence proved.