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

## Theorem

Let $x H$ denote the left coset of $H$ by $x$.

Then:

$x H = y H \iff x \in y H$

## 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:

$x H = y H$

$\Box$

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.

$\blacksquare$