Cosets

Suppose that a ∈ xH. Then a = xh1 for some h1 ∈ H. Therefore, since x = yh−1, you can write a in the form a = xh1 = yh−1h1. Since H is a subgroup, h−1h1 ∈ H from Theorem 20. Therefore a = y(h−1h1) ∈ yH, so xH ⊆ yH.