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 $-: R \times R \to R$ be the mapping defined by:
 * $\forall x, y \in R: \map - {x,y} = x-y$

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 $S$.

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

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

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