Math Help - Correct another congruence/mod proof, please!

1. Correct another congruence/mod proof, please!

Can somebody please check my work on this proof? I would greatly appreciate it!

Prove a ≡ b(mod m) if and only if there exists an integer k such that a = km+b.

(1)
If a ≡ b(mod m), then a = km+b, where k is an existing integer.
Proof: (⇒)Suppose a ≡ b(mod m), then m|a-b.
Thus a-b = km where k is an existing integer.
Therefore, by addition, a = km+b.(⇐)

(2)
If a =km+b and k is an existing integer, then a ≡ b(mod m).
Proof: Suppose a=km+b for an existing integer k.
Then a-b=km. Thus m|a-b.
Therefore a ≡ b(mod m).

Hence, a ≡ b(mod m) ⇐⇒ a=km+b for an existing integer k.

2. Re: Correct another congruence/mod proof, please!

looks fine to me