Negative of Subring is Negative of Ring

Theorem
Let $\struct {R, +, \circ}$ be an Ring.

For each $x \in R$ let $-x$ denote the ring negative of $x$ in $R$.

Let $\struct {S, +_{\restriction S}, \circ_{\restriction S}}$ be a subring of $R$.

For each $x \in S$ let $\mathbin \sim x$ denote the ring negative of $x$ in $S$.

Then:
 * $\forall x \in S: \mathbin \sim x = -x$

Proof
Let $\iota: S \to R$ be the inclusion mapping from $S$ to $R$.

By Inclusion Mapping on Subring is Homomorphism, $\iota$ is a ring homomorphism.

By Ring Homomorphism of Addition is Group Homomorphism, $\iota$ is a group homomorphism on ring addition $+$.

Let $x \in S$.

Then: