Left Cosets are Equal iff Element in Other Left Coset/Proof 2

Proof
Let $x \in y H$.

Then $x$ is of the form $y h_1$ for some $h_1 \in H$.

Thus every element of the form $x h_2 \in x H$ is of the form $y h_1 h_2$ for some $h_2 \in H$.

But:
 * $h_1 h_2 \in H$

and so:
 * $x h_2 \in y H$

So by definition of subset:
 * $x H \subseteq y H$

Let $x \in y H$ again.

Then $x$ is of the form $y h$ for some $h \in H$.

But then:
 * $y = x h^{-1} \in x H$

Thus every element of the form $y h_2 \in y H$ is of the form $x h^{-1} h_2 \in x H$.

Thus by definition of subset:
 * $y H \subseteq x H$

By definition of set equality:
 * $x H = y H$

Let $x H = y H$.

Then $x h_1 = y h_2$ for some $h_1, h_2 \in H$.

Hence:
 * $x = y h_2 h^{-1} \in y H$

The result follows.