![\[sita\] Paths $P$ and $P'$ is case (a).](sita.eps)

-   $x=a$ and $y \notin V(P')$:\
    In this case, we obtain a $4$-cycle $C=x,b,v,y,x$ such that the edge $xu$ exits $C$, see Figure \[sitb\]. Moreover, $P'$ is the only such path (otherwise we obtain two distinct $4$-cycles with two edges in common).

    ![\[sitb\] Paths $P$ and $P'$ in case (b).](sitb.eps)

-   $y=b$ and $x \notin V(P')$:\
    In this case, we obtain a $4$-cycle $C=u,x,y,a,u$ such that the edge $yv$ exits $C$. Moreover, $P'$ is the only such path (otherwise we obtain two distinct $4$-cycles with two edges in common).

Next we show that cases (a) and (b) or (a) and (c) can not happen simultaneously. Suppose that there is a path $P'=u,x,b,v$ of length $3$ between $u$ and $v$ such that $y\neq b$ (case (b)). Moreover, let $P''=u,a',b',v$ be another path between $u$ and $v$ such that $x \notin V(P'')$ and $y \notin V(P'')$ (case (a)).

-   If $b'=b$, then we obtain a $4$-cycle $x,y,v,b,x$ and a $6$-cycle $u,x,y,v,b,a',u$ with three common edges - a contradiction.

-   If $b'\neq b$, then we obtain a $6$-cycle $u,a',b',v,y,x,u$ and a $6$-cycle $u,a',b',v,b,x',u$ with three common edges - a contradiction.

Since we get a contradiction 