Subtraction of Subring is Subtraction of Ring

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

For each $x, y \in R$ let $x - y$ denote the subtraction of $x$ and $y$ in $R$.

Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$.

For each $x, y \in S$ let $x \sim y$ denote the subtraction of $x$ and $y$ in $S$.

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

Proof
Let $x, y \in S$.

Let $-x$ denote the ring negative of $x$ in $R$.

Let $\mathbin \sim x$ denote the ring negative of $x$ in $S$.

Then: