Is one of these incorrect?

Also, I often see in proofs that "x inZ" and later they say "but x is arbitrary" is it wrong to use an existential quantifier at the beginning, or is it implied?

1.

(Universal Quantifier) m inZis even <=> (existential quantifier) k in Z such that 2k = m

2. m inZis even <=> (existential quantifier) k inZsuch that 2k = m