Equivalence of Definitions of Lucas Numbers

Definition 1 implies Definition 2
Let $\sequence {L_n}$ be the sequence defined as in definition 1.

It follows from Lucas Number as Sum of Fibonacci Numbers that $\sequence {L_n}$ is the sequence defined as in definition 2.

Definition 2 implies Definition 1
Let $\sequence {L_n}$ be the sequence defined as in definition 2.

It follows from Lucas Number as Element of Recursive Sequence that $\sequence {L_n}$ is the sequence defined as in definition 1.