Vectors are Left Cancellable

Theorem
Let $\struct {\mathbf V, +, \circ}$ be a vector space over $\GF$, as defined by the vector space axioms.

Then every $\mathbf v \in \struct {\mathbf V, +}$ is left cancellable:


 * $\forall \mathbf a, \mathbf b, \mathbf c \in \mathbf V: \mathbf c + \mathbf a = \mathbf c + \mathbf b \implies \mathbf a = \mathbf b$

Proof
Utilizing the vector space axioms: