Vectors are Right Cancellable

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

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


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

Proof
Utilizing the vector space axioms:

Also see

 * Vectors are Left Cancellable