$ and $\bar{L_{\beta}}$ embeds in the automorphism group of $Q_{\beta}$, it follows from \cite{Winter} that $Q_{\beta}\cong 3^{1+4}_+$ and we have (ii) as a conclusion. If $G$ is obtained from a fusion system $\mathcal{F}$ satisfying \cref{HypFus}, then applying the results in \cite{Comp1}, $\mathcal{F}$ is isomorphic to the $3$-fusion system of $\mathrm{PSU}_6(2)$ or $\mathrm{PSU}_6(2).2$.
\end{proof}

We conclude this final section by summarizing what has been shown:

\begin{theorem}\label{Mainbodd}
Suppose that $\mathcal{A}=\mathcal{A}(G_\alpha, G_\beta, G_{\alpha,\beta})$ is an amalgam satisfying \cref{MainHyp}. If $Z_{\alpha'}\le Q_{\alpha}$, then one of the following holds:
\begin{enumerate}
\item $G$ has a weak BN-pair of rank $2$; or
\item $p=3$, $b=1$, $|S|\leq 3^7$ and the shapes of $L_{\alpha}$ and $L_{\beta}$ are known.
\end{enumerate}
\end{theorem}

We had this on our minds for 2 years since it was announced London would host the 2013 world triathlon finals. Not as exotic as our previous outings in Beijing, Eilat or Alanya but that would do fine as a more local event.
Competition was tough earlier in the season for qualification places as all the Brits were trying their luck as well but team Zoppi managed to book 2 spots for the sprint race with their respective federations.
After an early massage we set off for the capital for the season finale the car full of gear and kit. We reached Hyde Park by mid afternoon and took possession of our one room apartment. Trickiest part was to find a safe spot for the car for the next 3 days. Easier said than done; after 40 min I was still neither a permit holder nor able to fit 3 days in a 4 hour parking time allocation. Not impressed I was going back to the start when I saw the magic P sign in a blue background. A £72 victory!
Now onto serious things and the business of trying to register and get a feel for the venue. After 30 min walk across the park we found the venue but nothing amazing is to be found. It was hailed as venue with an Olympic feeling but that was nonsense. There was absolutely nothing to give a hint that Olympic events happened here a year ago. We found registration to be closed (2 hours earlier than planned, thanks) and it was raining by now, lovely. Time to head back the studio and get some Italian dinner with fellow competitors Sam and Hannah.
After a poor night sleep (noisy neighbours form above) we got up early to register and do a proper course reccie. It started raining a soon as set foot outside the door. We were warned we couldn’t go on the cycle course but nothing would stop us not to try. After a short ride in the park we rejoined the course loop. We managed to stay on the road for the half a lap until well rehearsed marshals stopped us, gesticulating in all directions. We tried going round but they were having none of it so had to stay on the pavement at snail pace. Not ideal and actually pretty useless, but by then I knew it was tough, twisty and turny with a very slippy road surface.
We next set to check out the transition area. Not impressed again. A field slightly uphill to accommodate a couple of thousand athletes. This will end up like a muddy cross country field as downpours are forecast for the next 24hours. Registration was quiet and quick: a plastic bag, Tshirt and energy sachets are about all you get for €230 these days in London. And they’d already run out of small sizes Tees on day 2 of a 5 day event. Not impressed number 4 or 5.
Soaked and cold we head back to base for breakfast before another trip to the race venue for the race briefing. Nothing useful there and we leave before the unavoidable load of silly questions by other competitors.  Lunch is followed by bike cleaning following the early morning filthy ride.
By late afternoon we had the bike ready and we made our way back to transition for racking. Luckily there was no queue at the gate so we were quickly allowed on the good to soft T area. Despite the late afternoon sun we knew forecast for the night and the next day was very wet. I’d been meaning to get a bike cover for the last two years…total fail. However I wasn’t the only one and noticed that many had yellow branded plastic covers. I race walk out of transition and towards the expo section and managed to secure the last two covers. Much better than cutting and taping through bin bags I took with me as an alternative.
Final checks around transition and it’s nearly time for dinner, the last one. Dinner (carbs, protein and veg) kit bag checks and sleep.
Race day. Up at 5:30 after another disrupted night (drunk cow pacing up and down he street effing and blinding for 2 hours). Feeling unusually nervous this time around. Breakfast as usual (scrambled eggs and plum tomatoes) and we were out of the door in no time. The ground is definitely wet. It was yet another 30 min walk back to transition. After 30 sec I popped the question: “You have everything? Your Helmet?”. “No” I heard in reply. Lucky day. A quick turn around and the said helmet is recovered. We are now truly on our way.
25 min later as we approach transition another panic sentence: “I forgot my wetsuit!”. Not good but not a disaster as we have two hours to wait between transition closing time and race start so plenty of time to go back (again). So the new plan is to do our transition setup quickly and go and get the wetsuit after that. Actually this was a blessing in disguise as some unnecessary gear could be returned to the apartment. Back on site with well over an hour spare. Then it’s the long wait, warming up and looking at other waves starting. Nerves go up one notch at this point so time to go in a quiet place and breathe. Oh yeah it started to drizzle by now as well.
8:40. Time to get the wetsuit on, drop the clothes at the bag drop area and head to assembly area. Wait. Go to holding area. Wait some more. After a selected few H&S warnings we are allowed on the start pontoon. Instructions are clear: walk to the end in a line, get your allocated spot and sit down before further instructions. Sitting on the blue carpet, everything feels very calm. A bit cold when the feet touch the water but nothing like the North Sea where it is instant freeze. Then it’s the one minute announcement before entering the water. Extremely quiet; the water is also calm (very flat too), a bit surreal. I could happily have walked away at this point.
In the water, one hand on the pontoon. Set. Go. The battle commences, squeezed between my left and right competitors, those guys are clearly worse than me at sighting. It’s a battle for the first 400 or 500m as everyone wants to get the best line (even those at the back). The swim is over and as I stand up on the ramp out I don’t feel too bad (no dizziness nor exhaustion). Did I go hard enough? Or did I leave too much for later? Doesn’t matter it’s too late to think about that. Time for the long run up to transition, unzipping the wetsuit, taking upper part, goggles and swim cap off. Into transition on the wet grass, soft to heavy by this point, mud between the toes.  On the bike and go. After the first few hundred metres visibility is reduced to a few tens of metres and I definitely have no useful brakes. I think it’s raining proper now. As usual I go hard on the first lap and nearly get caught on a couple of speed bumps. A few near misses on the bends and the first lap is over. I hear a few supported shouting my name I think I can even recognise the voices but no way I’m taking my eyes off the road or what I can see of it. The next two laps are more of the same but feeling less and less confident and speed is dropping.
Last corner before heading back towards transition. I can hardly see the cones or the signs on the sides of the path. I remove my feet of the shoes far too early and even apply the full brakes (or what I have left of them) 200m before the line I can’t see.  Another push on the pedals to gain momentum and the real dismount line appears. Off the bike (not my greatest skill) and up the muddy hill all the way round transition. Rack bike, helmet and glasses off, trainers on and off we go.
Legs feeling ok out of transition. The run route is nice. After the first couple of hundred metres with spectators the next kilometre is much quieter. As you turn back onto the bridge toward the end of the first lap there are big crowds and you get a definitive lift. Much needed as I have been caught by a kiwi who hasn’t dropped me and I’m feeling confident I can hold onto. The third km is hard