Ordered_Group_Equivalences

Let $\left({S, \circ, \preceq}\right)$ be an ordered group whose identity is $e$. Let $x, y, z, \in S$. Then the following are all equivalent $(1): \quad x \prec y$ $(2): \quad x \circ z \prec y \circ z$ $(3): \quad z \circ x \prec z \circ y$ $(4): \quad y^{-1} \prec x^{-1}$ $(5): \quad e \prec y \circ x^{-1}$ $(6): \quad e \prec x^{-1} \circ y$