Head across the bridge and you'll soon see Mathias. Of course Himiko will get in the way and allow Mathias to escape.
After the cutscene go to the opening on the left side and turn right. Fire a rope arrow at the climbable part of the wall and head up using the zip line.
Scale the wall and jump onto the platform at your right side. Now jump towards the next platform and attach your climbing axe to the ice.
Climb up the ice and get on top of it. Then head back inside the building via the broken window. Stand on the left side of the gate and fire a rope arrow at the bell. The wind will then push it and destroy the wood keeping the door shut.
Enter the next area and go to the left side. Use the fallen staircase to make your way to the upper floor. Stand near wooden pillar and fire a rope arrow at the bell on the higher floor and use the ascender to get there.
The building will start to collapse and you'll fall down. Press right to make Lara lean towards that side and grab the wood that's sticking out of the floor.
Now make your way to the right side and jump towards the platform on the opposite side.
Now follow the path until you get to the fallen stairs. Jump towards it and it will fall down. Press E to attach your climbing axe to the rocks and make your way to the right side then up near the windows.
Jump up towards the decorative piece of the wall and use it as a ladder to get higher. Enter the tower through the broken section of the wall.
Then left and the thunders will start chopping off sections of the ice. Use your rope arrow to create a zip line for you to use.
Once you're on the ice make your way to the left side and jump towards the ice on the lower left. Press E to attach your climbing axe. Then drop down again with shift and press E to attach the climbing axe on the lower portion of the ice.
Now make your way to the right side and up the ledge that's on the left side of the large statue.
Head through the door and use the pole to boost you towards the other side. Continue climbing up the wall until it starts to break down and you can't proceed any further.
When that happens shimmy over to the right and continue climbing on that side and continue climbing up past the ice wall. A cutscene will play after you reach the top.
After the cutscene climb down the platform and hide behind something. Once the wind dies down the Soalrii will attack you again. All of them are armored at this point. Start shooting their heads with the bow, the arrows should go through their helmets if you took the penetrating arrow upgrade.
Once you've killed them go towards the flaming barricade and the wind will start to pick up again causing a large pillar to crash down. Turn right then use that wall to climb up.
Climb down the stairs and more Salarii will attack you. Hide behind cover again and take them out.
Go towards the flaming barricade again after you've eliminated the Solarii and another path will open up. Jump on top of the small platform that was left behind then jump towards the other side, use your climbing axe to climb up the opposite wall.
You're going to go against the big guy. Finally. Now equip your shotgun and wait for him to swing his club at you. Dodge it and you'll see that his back is completely unprotected. Shoot his back with the shotgun.
After about 4-5 blasts from the shotgun he'll get on his knee and a melee icon will appear on his head. Get close to him and smash his face with your axe.
Once he recovers he'll have some back up archers. Luckily for you this guy doesn't care about his comrades. Stick close to him and get the guy to swing his club at the archers, dodge it then shoot his back.
After you bring him down a second time you'll be able to see his face after smashing it with the axe a second time. At this point you don't have to dodge his attack. Just shoot at his head until he falls one last time.
And finish him off with your climbing axe.
Now climb up the stairs and turn left. Head for the glowing light at the end of the bridge. It's time to bring down Mathias.
As you approach Sam, Mathias will shoot your bow right off of your hand. Lara will take out her pistol and you have to start shooting at Mathias whether you hit him or not.
He'll attempt to tackle you. Press F at the right time to bury the axe into his shoulder. He'll succeed in tackling you though.
And he'll attempt to strangle you with the climbing axe. The screen says to mash E but if you press it too fast like for example EEEEEEEEEE, he will most definitely kill you. Instead of pressing too fast try to press a little slowly (E E E E) with small pauses in between and you should be able to win the power struggle.
After winning the power struggle Lara will grab his pistol or rather the other half of Roth's pistols and shoot him.
Lara will then take on her signature weapon and you have to fire both pistols at him to kill him.