Wait for the right moment to make a jump towards the swinging platform. Once you're there, wait for it to get closer to the other side of the building. Make a double jump in order to use the grapple once more (#1). Be careful! Don't make any jumps just yet. You will have to turn right. As a result, you'll see a new platform ahead of you (#2). Start swinging towards it and then make a new jump.
Once again you will have to use the grapple to get the metal platform swinging. After that, make a jump towards it (#1). Now you will have to repeat these actions with the second platform. Once you're there, make a double jump in order to use the grapple. Turn right and press the E key. Lara must climb up a little (#2). Now you will have to reach the roof area (start swinging).
Ready your guns! Once again you will have to deal with a large group of enemy soldiers. Be careful. There are some attack dogs with them. You will have to kill them almost instantly, because they pose a serious threat. I would also recommend jumping a lot. You might want to consider shooting at the blue lanterns (#1), however this is not necessary. Don't forget about the ammunition and medkits! Once you're done with this problem, head on to the only visible door (#2). Press the E key in order to open it.
You will find yourself on a small staircase. Your current destination is on the upper floor, so you must go there now. You will come across a new door here (#1). Open it. There are a lot of enemy soldiers in the new room (#2). You will have to kill them all. Don't forget to dodge most of their bullets. I would also recommend that you take out the nearby snipers quickly. They are hiding on the right platform. Once you've killed everyone, grab all of the necessary objects. Now you will have to solve a simple puzzle.
Head on to the big screen. It's located in the central part of the room. Once again, you will have to use the grapple in order to pull it towards Lara (#1). The change won't be significant, but it'll do. Now you have to ready one your guns. You will have to enter the Accurate Aim mode and eliminate both of the highlighted objects (#2). Thanks to that, the whole screen will come crashing down.
Once you're done with that problem, make a jump towards the screen (you will have to be standing on a nearby stairs). Lara is going to be attacked by some additional enemy units (#1). You might also consider eliminating these thugs from a distance, however this is not necessary. Use the nearby door. A short cut-scene will appear on your screen (#2). As you've probably suspected, you will have to reach the top level of the new construction, so there's a long and difficult climb ahead of you...
Start off by using a nearby pipe. You shouldn't have any problems locating it. Once you're at the top, make a single jump towards the new construction (#1). Start moving towards the right edge of the screen. The objective is to reach (by making a single jump) the top of this platform. Now you will have to use the grapple. Thanks to that, you will be able to lower a huge metal construction (#2).
You shouldn't have any problems with some of the upcoming jumps. Once you've reached a ladder, start moving towards the upper part of the construction. Use the grapple in order to pull the huge object down. Make a first jump towards the metal pole (#1). The second one will allow you to reach the main platform. There's a small pipe here. Use it (#1). Start moving upwards. Once you're there, make a jump towards a new platform.
You should be able to notice a new pipe on your right. Start running towards it. Once you've made a successful jump, climb up a little (#1). Now you will have to reach a metal pole (#2). Thanks to that, you will be able to lower a new platform. Turn around and start swinging on the pole. You will have to make a jump towards the pipe (the same one!). Continue moving upwards.