ISL 1996 C7

let $V$ be a finitive set and $g$ and $f$ be two injective surjective functions from $V$ to $V$ .let $T$ and $S$ be two sets such that they are defined as following" $S = \{w \in V: f(f(w)) = g(g(w))\}$ $T = \{w \in V: f(g(w)) = g(f(w))\}$ we know that $S \cup T = V$ , prove: for each $w \in V : f(w) \in S$ if and only if $g(w) \in S$