Triest can't support Venezia and Battle in Venezia is draw.
Is this right? I confuse because The Italian Army in Tyrolia can't Attack Trieste and not win, return to hold Tyrolia before attack from Vienna of Russian Army?
You can see map:
Before Orders: http://i48.photobucket.com/albums/f235/Argon18/Winter1906_zpsbae11fea.jpg
After Orders:
http://i48.photobucket.com/albums/f235/Argon18/Spring1907_zpsaa1ae411.jpg

Support is cut even.if the unit is;'t dislodged except under some very specific orders involving either a unit trying to cut support against itself or supporting a move against a convoying fleet directed at itself.

R Vie-Tyo
I Tyo-Tri
I Ven H
A Tri SM Apu-Ven
T Apu-Ven

Tyo-Tri cuts Tri support.
Apu-Ven bounces.
Vie-Tyo bounces.

He adjudicated it perfectly if that is what he had happening.

Yes, the support is cut and everything stays where it started. You're thinking about it in slightly the wrong way - the army doesn't so much go to Trieste, fight a battle there, and then come back to Tyrolia. It's more like it attempts to move to Trieste, but gets stopped at the border, and therefore never actually stops occupying Tyrolia.

. . . yet the the march to the border forces the Austro-Hungarian forces in Trieste to defend against Tyrolia, so the Austro-Hungarian forces can't also help Turkey out with Ven.

yes, this is an example of a unit affecting two provinces at the same time.

In this case cancelling vienna's attempted move to tyrolia, AND trieste's support order.

Note the first exception to the rule on cutting support is that a unit can't support aimed at it. So if Trieste's order had been 'support Vienna move to Tyrolia' then the Russian attack would have worked.

Sephiroth: It can be called called a 'tap' and/or a 'bounce'. Italy purposely 'tapped' Tri to make sure F Tri couldn't order a support. Of course, A Tyr 'bounced' back as well and as a result 'bounced' Russia out of Tyr too.

I'm not sure any of this makes sense. You can read a copy of the rulebook here to get all the details and explanations:
http://www.stabbeurfou.org/Regles.php
(This is the best copy I've found, but I haven't been able to download the PDF recently)

If you are ever confused, this is a perfect tool with which too test out different scenarios without having to download anything:
http://gamesbyemail.com/Games/Politics/Judge
To clear the map and build your own scenario, click the "Clear board" hyperlink under "Other:" and then drag and drop the units from the white box. To come back to a scenario you've created, copy the Permalink the the right of the board.

orathaic and Sephiroth: Regarding orathaic's example, note that the Austro-Hungarian fleet would actually have to be an army. A fleet can't order any support into a territory it can't move to..

Thanks all. I understood.

@JECE, thanks, i stand corrected (that's what i get for not looking at the map)
and the rules download for me: www.wizards.com/avalonhill/rules/diplomacy.pdf‎

So all that just to confirm what I already told him... It was adjudicated correctly.