Order_of_Squares_in_Ordered_Ring_v3

Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $x, y \in \struct {R, +, \circ, \le}$ such that $0_R \le x, y$. Then $x \le y \implies x \circ x \le y \circ y$ When $R$ is one of the standard sets of numbers, that is $\Z, \Q, \R$, then this translates into If $x, y$ are positive then $x \le y \implies x^2 \le y^2$.