Proposition 10

Let x, y and z be three elements of an ordered group ; for x – z and y – z to be coprime, it is necessary and sufficient that z = inf (x, y).

Indeed the relations $z = \inf(x, y)$ and $0 = \inf(x - z, y - z)$ are equivalent.

Proposition 10 (DIV). — Let a, b and c be three elements of K with c $\neq 0$; for the quotients $ac^{-1}$ and $bc^{-1}$ to be coprime, it is necessary and sufficient that c be a gcd of a and b.