ISL 1970 Problem 28

A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled: $(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$ . $(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$ . $(\text{iii})$ $\forall \{u,v\}\in G$ , there exists an element $x\in G$ such that $u\circ x = v$ , and an element $y\in G$ such that $y\circ u = v$ .

Let $K$ be a set of all real numbers greater than $1$ . On $K$ is defined an operation by $a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$ . Prove that $K$ is a Group.