# 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:

 $\displaystyle x \sim y$ $=$ $\displaystyle x \mathbin {+ {\restriction_S} } \paren {\mathbin \sim y}$ Definition of Ring Subtraction $\displaystyle$ $=$ $\displaystyle x + \paren {\mathbin \sim y}$ Definition of Addition on Subring $\displaystyle$ $=$ $\displaystyle x + \paren {-y}$ Negative of Subring is Negative of Ring $\displaystyle$ $=$ $\displaystyle x - y$ Definition of Ring Subtraction

$\blacksquare$