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 $-: R \to R$ be the mapping defined by:
 * $\forall x \in R: \map - x = -x$

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 $S$.

Let $\sim: S \to S$ be the mapping defined by:
 * $\forall x \in S: \map {\sim} x = \mathbin \sim x$

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

Equivalently:
 * $\mathbin \sim {} = -_{\restriction S}$